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A BRIEF INTRODUCTION 


TO 


RHE SINEINEPESIMAL GCALCULUS 


A BRIEF INTRODUCTION 


PEC. . YWealor, »/409 


INFINITESIMAL CALCULUS 


DESIGNED ESPECIALLY TO AID IN READING 
MATHEMATICAL ECONOMICS AND 
STATISTICS 


BY 


DR VON Giles Ee Rebar t), 


PROFESSOR OF POLITICAL ECONOMY IN YALE UNIVERSITY 
Co-AUTHOR OF PHILLIPS’S AND FISHER’S ‘“‘ ELEMENTS OF GEOMETRY” 


THIRD EDITION 


Nets Bork 
THE MACMILLAN COMPANY 
LONDON: MACMILLAN & CO., Lt. 
1906 


All rights reserved 


J. a 


” 


Le 


—— 


CopyrIGHT, 1897, 


By THE MACMILLAN COMPANY. 


Set up and electrotyped 1897. Reprinted April, 1900; 
July, 19013 February, 1904; March, 1906. 


Norioood JPress ’ 
J. S. Cushing & Co. — Berwick & Smith 
Norwood Mass. U.S.A. 


PRE eee 


Cnet ole. 


) Tuis little volume contains the substance of lectures by 
‘which I have been accustomed to introduce the more 
“advanced of my students to a course in modern economic 
“theory. I could find no text-book sufficiently brief for my 
purpose, nor one which distributed the emphasis in the 
desired manner. My object, however, in preparing my 
notes for publication has not been principally to provide a 
book for classroom use. It must be admitted that very few 
teachers of Economics as yet desire to address their stu- 


dents in the mathematical tongue. I have had in mind not 


so much the classroom as the study. Teachers and students 


Yr 


alike, however little they care about the mathematical 
~ medium for their own ideas, are growing to feel the need of 
it in order to understand the ideas of others. I have fre- 


At ch dh 


a 


"7 


“~quently received inquiries, as doubtless have other teachers, 
«for some book which would enable a person without’ special 
mathematical training or aptitude to understand the works 
of Jevons, Walras, Marshall, or Pareto, or the mathematical 
articles constantly appearing in the Hconomic Journal, the 
f Journal of the Royal Statistical Society, the Giornale degh 


Economist, and elsewhere. It is such a book that I have 


Vy Our “~ an 


tried to write. 


vi PREFACE 


The immediate occasion for its publication is the appear- 
ance in English of Cournot’s Pyincipes mathématiques de la 
théorie des richesses, in Professor Ashley’s series of “ Eco- 
nomic Classics.” The “ non-mathematical” reader can 
only expect to understand the general trend of reasoning in 
this masterly little memoir. If he finds it as stimulating as 
most readers have, he will want to comprehend its notation 
and processes in detail. 

I have tried in some measure to meet the varying needs 
of different readers by using two sorts of type. If desired, 
most of the fine print may be omitted on first reading, and 
all on second. The reader is, however, advised not to pass 
over all of the examples. 

Although intended primarily for economic students, the 
book is equally adapted to the use of those who wish a short 
course in “ The Calculus ” as a matter of general education. 
I therefore venture the hope that teachers of mathematics 
may find it useful as a text-book in courses planned espe- 
cially for the “ general student.”” I have long been of the 
opinion that the fundamental conceptions and processes of 
the Infinitesimal Calculus are of greater educational value 
than those of Analytical Geometry or Trigonometry, which 
at present find a conspicuous place in our school and college 
curricula. Moreover, they are almost as easily learned, and 
far less easily forgotten. 

IRVING FISHER. 

NEW HAVEN, September, 1897. 


Beier TOMI E THIRD EDITION 


In the present edition have been incorporated several 
changes and additions originally prepared for the German 
translation of 1904 and for a Japanese translation in prep- 
aration. 

A preliminary statement of the concepts of limits and 
several new examples have also been inserted. 


IRVING FISHER. 


November, 1905. 


+i 


CONSE NEES 


PAGE 


INTRODUCTION . s : R ‘= : - : : ; xi 
CHAPTER I 

THE GENERAL METHOD OF DIFFERENTIATION . : : 5 I 
CHAPTER. II 

GENERAL THEOREMS OF DIFFERENTIATION Ms ; A 2 16 


CHAPTER III 


DIFFERENTIATION OF THE ELEMENTARY FUNCTIONS . . ye isete, 
CHAPTER IV 

SUCCESSIVE DIFFERENTIATION — MAXIMA AND MINIMA . ee RY 
CHAPTER V 

TAYLOR’S THEOREM . - : ° ° ° . . - 49 


CHAPTER VI 


INTEGRAL CALCULUS . : : . . . . . ome S 7 


APPENDIX 
FUNCTIONS OF MORE THAN ONE VARIABLE - ; ay 


1X, 


INTRODUCTION 


TuE reader of the following book should be familiar with 
ordinary algebraic operations and with the concepts of vari- 
ation and limits, a brief statement of which is here appended. 

Continuous Variation. — Suppose the line @d to represent 
all possible magnitudes between — a and +4; suppose om 
to represent one magnitude between —a@ and +4; this 


magnitude is said to vary continuously when it increases or 


—a 0 m mM, Mz Me +3 
oe cae in anal scale en LE el de ts ead eed EE MSS Mie 
HiGaats 


decreases in such a manner that 7 may occupy any position 
whatever between — a and + 4. 

Limits. — If we conceive om to have an infinite succession 
of magnitudes such that # may occupy the positions 7, mz, 
ms, etc., making the w/t#maze difference between of and om 
less than any assignable positive quantity, then 077 1s a vart- 
able and o6 is its Lmtt. 

It is clear, then, that the difference between the limit od 
and the variable om is another variable magnitude whose 
limit is zero. A variable, with a limit zero, is called an 


infinitesimal. 
xi 


xii INTRODUCTION 


Application to Infinite Series. — In a converging infinite 
series, the sum of each successive term and those preceding 
approaches a magnitude understood to be designated by 
the series. ‘This magnitude is called the ‘sum’ of the series. 

Thus, the repeating decimal .666---, 


6 
Or, ele Zh Se se, 


IO 


means a series of successive magnitudes, viz.: 


(a) , which is less than 2 


(6 he -—~ Sea =o} which is less than 2, but more nearly approxi- 


mates 2 than (a). 
ON == oi ae 3, which is less than’, but more nearly 


rosin than wh 


(2) Ba “8 2 —» Which is less than 3, but more 


nearly approximates 2 than (¢). 

Thus, as the number of terms of the series is increased, 
the sum of the terms remains always less than 2, but approx- 
‘mates ultimately as nearly 2 as may be desired, z.e. converges 
towards 2. We therefore, by convention, speak of 2 as the 


‘sum,’ or limit, of this infinite series. 


THEOREMS. 
1. Zhe limit of the sum of two different variables (which 
approach limits) ts the sum of the limits of those variables. 


INTRODUCTION xiii 


2. The limit of the difference of two different variables 
(which approach limits) ts the difference of the limits of those 
variables. 

3. Lhe mit of the product of two different variables 
(which approach limits) ts the product of the limits of those 
variables. 

4. Zhe limit of the quotient of two different variables 
(which approach limits) ts the quotient of the limits of those 


variables. 


INFINITESIMAL CALCULUS 


CHAPTER I 
THE GENERAL METHOD OF DIFFERENTIATION 


1. The Infinitesimal Calculus treats of the ultimate ratios 
of vanishing quantities. This definition, however, can only 
become intelligible after some actual acquaintance with 
“ultimate ratios.” 


2. The conception of a limiting or ultimate ratio is funda- 
mental in many familiar relations. It is impossible, without 
it, to obtain a clear notion of what is the velocity of a body 
at an instant. ‘The average velocity of the body during a 
period of time may readily be defined as the quotient of the 
space traversed during that period divided by the time of 
traversing it. Ifa steamer crosses the Atlantic (3000 miles) 
in 6 days, we may say that the average speed is 3000 + 6, 
or 500, miles per day. But this does not tell us the speed at 
various points in the voyage, under head winds, storms, or 
other conditions, favorable or unfavorable. What, for 
instance, was the speed at noon of the third day out? We 
may obtain a first approximation to the desired result by 
taking the average speed for a short time after the given 
instant ; that is, taking the ratio of the distance traversed 

T 


2 INFINITESIMAL CALCULUS 


during (say) the following hour to the time of traversing it, 
which is 5; of a day. If this distance be 20 miles, we obtain 
20 + >1,, or 480 miles per day, as the average speed during 
that hour. For a second approximation we take a minute 
instead of an hour ; for a third, a second instead of a minute, 
and so on. ‘The ratio of the space traversed to the time of 
traversing it becomes closer and closer to the true speed. 
Though both the time and space approach zero as limit, 
their vat#o does not. The limit which this za#o approaches, 
or the w/timaze ratio of the distance traversed to the time of 
traversing it when both distance and time vanish, is the pre- 
cise speed at che instant. 


3. Let us apply this method of obtaining velocity to 
bodies falling in a vacuum. We know from experience that 
the distance fallen equals sixteen times the square of the 
time of falling, ze. s= 167%, where s is the distance fallen 
from rest (measured in feet), and ¢ is the time of falling (in 
seconds). Consider the body at some particular instant, ¢ 
being the time to this particular point and s the distance. 
Suppose we wait until the time has increased by a small 
increment A¢, during which the body increases its distance 
from the starting-point, s, by the small increment As. Since 
the above formula holds true of a/Z points, it holds true now, 
when the time is ¢+A¢, and the distance is s+As. That is, 


s+As= 16(¢+ As)’. 
This gives . 
s+As= 162+ 322- At+ 16(A?)’, 


But 5 aoe e 
Subtracting, we have 
As = 327+ At+ 16(A?)’, 


GENERAL METHOD OF DIFFERENTIATION 3 


whence 7 = 324+ 16 Ad. (1) 


This is the average velocity during the small interval Az 


Thus, if A¢ = } second and ¢ be 5 seconds, the average speed of the 
body during that half second (viz., the one beginning 5 seconds from 
rest) is 32X5+16X 4, or 168 feet per second. If we take ;4, of a sec- 
ond instead of 3, we have 32 x 5 + 16 X 745, or 160.1 feet per second. 


Thus, by taking AZ smaller and smaller, we obtain the 


average velocity a for a smaller and smaller interval of time 


immediately after the completion of the fifth second. The 


limit which ~ approaches, as A¢ approaches zero as its 


limit, is called the velocity at the very zzs¢an¢ of completing 
the fifth second. 

Its value is exactly 160, as is evident from the right-hand 
member of equation (1), which approaches as its limit (as ¢ 
is 5 and A¢ approaches zero), 


32 X 5 +16 X 0, or 160. 
In general, to express the limit of both sides of equation 
(1) when Af approaches zero, we write 


hives Re. 
A¢ 


4. The student will observe that, as A¢ approaches zero, 
As also approaches zero, since a body cannot pass over any 
distance in no time. He must be warned, however, against 


expressing the limit of cs by of which, of course, is quite 
fe) 


indeterminate. 
But in spite of the fact that the vato of these “mits of As 
and A¢ is indeterminate, the Zmz¢ of the rato of As and At 


4 INFINITESIMAL CALCULUS 


may be entirely determinate. It is only with this latter con- 


ception, viz. the limit of alah or lim rad that the student has 
Az Af 
to deal. 
The limit of the ratio of the vanishing quantities As and 


4é, or lim x, is called the “ derivative” of s with respect to 
¢; because, from s= 167? we derive lim 7 = 32 f. 


In fact, we may speak of either member of the latter of 
these two equations as the derivative of either member of the 
former equation. For instance, 32 ¢ is the derivative of 16 7”. 


5. Other names and notations are also used. Thus in- 
stead of lim = it is usual to employ the shorter symbol o 


In this expression @s and dare called aiferentials of s and ¢, 
just as As and AZ are called zmcrements of s and 4 But they 
are not zeros. They have no definite value individually. We 
may select any value we please for one of them. But when 
this one is fixed, the other is also, since the two must be kept 


in a ratio equal to lim ~ We say therefore that the differ- 


entials @s and d@¢ are any two quantities which bear to each 
other the ratio which is the limit of the ratio between As 
and Ad. 
yA S ads . ON a a 
Other names for lim cy or 7} besides “ derivative,” are 


“ differential quotient’ and “ differential coefficient.” 


6. In the particular case considered above, the differ- 
ential quotient is a velocity and may be denoted by z. 
Equation:(2) thus becomes* v= 324. 

* If distance be measured in centimetres instead of in feet, we should 


have v= 9804, and in general v = g#, where g is a constant depending 
for its numerical value on the units chosen for measuring space and time. 


GENERAL METHOD OF DIFFERENTIATION 5 


Velocity at a point may now be defined as the ultimate 
ratio of the space traversed just after passing the point to the 
time of traversing it when the space and time approach zero 
as limit. 


7. EXAMPLES. 

1. What is the velocity of a body which has fallen 10 seconds ? 
100 seconds ? 14 seconds ? 

2. What is the velocity of a body which has fallen 16 feet ? 

HINT. — First find how many seconds it has fallen by using s=16 2?. 


3. What is the velocity of a body which has fallen 64 feet ? 4 feet ? 
I foot ? 2 feet? 


4. It being known that a body, falling not from rest, but with an 
initial velocity of 5 feet per second, obeys the law 
$= 1627+ 54, (1) 
what will be its velocity at the end of any time 7? 


HINT. — Let ¢ receive an increment A¥, causing s to increase by As, 


so that 
s+ As = 16(¢ + Af)? + 5(¢ +4 Ad). (2) 


Subtract (1) from (2), divide by Av and then reduce A and As to zero. 
a ae 

A . ] i 2 i e 

72s. lim ae 322+5 


5. What will be the velocity at the end of 1oseconds? At the end 
of 69 feet ? 


6. It being known that a body falling with an initial velocity of z 
obeys the law s=4,¢* + wt, what will be its velocity at the end of 
bme.22. When 237 


8. When one quantity depends upon another, the first is 
said to be a function of the second. A change in the second 
is in general accompanied by a change in the first. In each 
case the limits, within which the function relation exists, 
should be specified. 


6 INFINITESIMAL CALCULUS 


~ 


Thus the distance a body falls from rest is a function of the time of 
falling, for how far the body falls depends on how long it has fallen; 
the demand for an article is a function of its price, for if the price 
changes the demand changes; if y = x?, then y is a function of x, for 
a variation in the magnitude of x necessitates also a variation in the 
magnitude of y. 


g. When one quantity is a function of another, the latter 
is called the zadependent variable, and the former the de- 
pendent variable. 

The distinction between the independent and the depend- 
ent variable is only for convenience of expression. The 
two may be interchanged. 

Thus, as the distance of a falling body from the starting-point 
changes, there is also a change in the time it has taken. Hence we 
may say that ‘time of falling” is a function of “ distance fallen.” Simi- 
larly price may be regarded as a function of demand. Again, y = x? 
may be written « = Vy, thus making x a function of y. The idea of 


functional dependence is therefore quite different from that of causal 
dependence. Functional dependence is a #zfua/ relation. 


In the example of falling bodies s was a function of 4 and 
what we accomplished was to find the differential quotient 
or derivative of that function. ‘The derivative in this case 
was a velocity. In general the process of finding the differ- 
‘ential quotient of any given function is called a@ferentiaton, 
and is the subject matter of the Differential Calculus, one 
of the two branches into which the Infinitesimal Calculus is 
divided. The Differential Calculus will occupy us in the 
first five chapters of this book. 


to. A second important application of the idea of a differ- 
ential quotient of a function is to the ¢angential direction of a 
curve at any point on it. The Calculus enables us to conceive 
in the most general manner of a tangent to a curve. The 


GENERAL METHOD OF DIFFERENTIATION “if 


student should observe that the usual definition of a tangent 
to a circle will not apply to any and all curves. A straight 
line may have only one point in common with a curve and 
yet cut it and not be tangent. 


11. Let AS be a curve whose equation is 
yoert+5x—x*. (1) 
That is, for azy point P upon it, the “ ordinate,” y (or dis- 


tance, 4, from that point to the horizontal axis), is related 
pi 


HiGa tT. 


~ to the “abscissa,” x (or distance, OA, from the vertical axis), 
in the manner expressed by (1). 4 isa function of O4 ; 
i.e. the height, PA, of any point P on the curve depends 
upon its distance, OA, from the vertical axis. 
What is the direction of the curve at the point P? The 
direction from the point P to another point 7’ is the direc- 
tion of the secant line Q'PP'. The point /' has for abscissa, 


8 INFINITESIMAL CALCULUS 


x + Ax, and for ordinate, y+ Ay. Since the relation (1) 
holds true of all points on the curve, it holds true of 7’. 


Hence y+ Ay= Ob ats (x fh Ax) wes («+ Avi 


or ytAy=1+5x%+ 5 Ax — 2x? — 2x Ax — (Ax)? 
Subtracting y=tt+5x—2, 

we have Ay = 5 Ax — 2x Ax — (Ax)’, 

whence aoe 5 —2x—Ax. 


Ax 


We may pause here a moment to see what this result 
Ay Ge i 73 ” : ! ! 
means. —— or ——is the “slope” of the line O'PP’. That 
Areas 
is, it is the rate at which a point moving from Q! toward ?’ 
rises in proportion to its horizontal progress. It is the same 
sort of magnitude as that referred to as the “ grade”’ of an 


uphill road which rises “so many feet to the mile (hori- 


zontally).” If .  — +, Q'PP' rises one foot in every ten 
Axe FLO 
horizontally. The “slope” of a line shows its direction. 


The equation mai 
Ax 


secant line Q!/P/' is to be found by taking 5 and subtracting, first, 
two times the number of units in O4 and then the number of units 
in AZ. For instance, if O14 = 2 and 48 = 3, then 


5 — 2x — Ax shows that the “slope” of the 


bY _ i" PR ee 
x =5-2x2-$=4; 


z.e. the secant slopes 1 fot up for every 2 feet sidewise. 
12. But we have not yet reached the tangent at P. Let 


the point P’ be gradually shifted along the curve toward P 
until it ultimately coincides. The secant Q'/’ will gradually 


GENERAL METHOD OF DIFFERENTIATION 7 


change its direction and approach a limiting position QP. 
This 4miting position we call the tangent. Its slope is 


ay =5 — 2%. 
ax 
Thus, if x(ie. OA) is 2, oy 1. That is, QP is inclined at 45°. 
, te 
Tix is 4, a =— 3; 2.e. the curve slopes down, not up. 
P 


Fic. 2.-— A, positive slope; B, zero slope; C, negative slope. 


EXAMPLES. 


1. What is the slope of the tangent to the above curve at the point 
whose abscissa is 1? O? 24? What does the answer to the last 
mean? 3? What does this mean? 6? —1? 


2. Derive the formula for the slope of the tangent to the curve 


yo=rtitunt x 


13. To construct a tangent at P, all we need to do is to 
draw a line through P with the required slope. ‘Thus, if we 
wish the tangent to the point whose abscissa is 1, we find 
from the above formula that its slope is 3. We therefore 
lay off a horizontal line ZA7 (Fig. 1) equal to any length vx, 
and at its extremity erect a vertical, AZ7/V, equal to three times 
as much, or dy. Draw ZV; this has the required direction. 
Then through / draw a line parallel to ZV. This will be 
the tangent. 

We may also call PC, dx and P''C, dy; for, by Sec. 5, dx 
and dy are simply any two magnitudes having a ratio equal 


ie A oak ie 
to the limit of mA when Ax approaches zero as its limit. 


The problem of drawing a tangent and calculating its slope was one 
of the chief problems which gave rise to the discovery of the Calculus. 


10 INFINITESIMAL CALCULUS 


14. It is evident that we could approach P from the left as well 
as from the right. We should, however, reach the same limiting posi- 
tion unless there should be an angle in the curve at the point / as in 
Fig. 3. In this case, the progressive (PA’) and regressive (///) tan- 
gents do not cuincide. 


Such peculiar points are not considered in this little treatise. All 
the functions are such that, for the values of the independent variable 
which are considered, the progressive and regressive derivatives are 
identical. The curves considered are all ‘‘smooth,” that is, have no 
angles or sudden changes in direction. In many applications of the 
Calculus, such as to statistical or economic diagrams, it is often con- 


venient first to smooth out the curves considered. When we want to’ 


see from a plot of the population what is the general rate of increase, 
we draw a tangent not to the plot of the actua/ figures, but to a swztoo/h 
curve coinciding as nearly as possible with the plot. 

The student will be able to satisfy himself in every particular case 
to be considered that the progressive and regressive derivatives are 
identical. 

Thus, for s=16 2? in section 3, let ¢ receive a decrement A't, causing 
s to have a decrement A/s. Then 


5 — A's = 16(¢— A!Z)2, 
Expanding, subtracting, and dividing as before, we obtain 


A's 


ay ote ! 
ia. 327—16A'4 
which reduces at the limit to 
a's 
Pai 324, as before. — 


Indeed, we assume in general, that it is physically impossible for 
a body to change its velocity per saltum. NWHence the definition of 


GENERAL METHOD OF DIFFERENTIATION 11 


velocity given in section 6 is equivalent to the following alternative 
definition ;: the ultimate ratio of the space traversed just defore reaching 
the point to the time of traversing it when the space and time ap- 
proach zero as limit. 

We shall, therefore, henceforth treat only of functions whose deriva- 
tives are continuous and which are themselves continuous, within the 
limits considered, that is, which in changing from one value to another, 
pass continuously through all intermediate values. 


15. We have seen that the conception of an ultimate ratio 
clears up the notion of velocity in mechanics and tangential 
slope in geometry. It is also applicable to much else in 
both these sciences as well as in all mathematical sciences. 
Momentum, acceleration, force, horsepower, density, curva- 
ture, marginal utility, marginal cost, elasticity of demand, 
birth rate, “ force of mortality,” are all examples. 

The conception of an ultimate ratio or of the derivative of 
a function is not dependent, however, on any special applica- 
tion. It is purely an abstract idea of number. 


16. Thus let two variables x and y fulfil the equation 
y= x", 
where # is a constant and a positive integer. We may 


obtain the differential quotient e for any particular value 


of x, as follows: - 


Let « receive an increment Ax producing an increment of 
y denoted by Ay. Then, by the binomial theorem, 


y+ Ay= (x + Ax)", 
= 0" + nx” Ax + n(n —1) Catan )4 
eer 
= x" + nx” Ax + Ax? (++), 


12 INFINITESIMAL CALCULUS 


Subtracting Y= x", 
we have Ay = nx" * Ax + (Ax)?(-++). 
Whence SY = nat) + ber (+), 

Ax 


where the parenthesis is evidently a finite quantity and re- 
mains finite after Ax becomes zero. Hence, when Ax 
becomes zero, the term Ax(--+) becomes zero, and the 
equation becomes, 

DY x age, 

ax 


17. This is the first and most important specific formula 
which we have reached for the derivative of a function. It 
states that, to obtain the derivative of x", a power of x, we 
need only reduce the exponent by unity and use the old 
exponent for coefficient. 


»Thus the derivative of «3 is 3%, When x passes through the value 
2, 3x? becomes 12; that is, vy, or x*, is increasing 12 times as fast as x. 


ay . 
ms the rate at which y increases compared with the rate we make x 
7 


increase. If y denotes the distance of a moving body from the start- 


ing-point, and x denotes the time it has moved, a or 3.x”, expresses 


its veloctty, Again, if « and y are the “codrdinates” (z.e. the ‘ab- 
scissa”’ and “ ordinate”) of a curve whose equation is y = x3, then 
3 x? isits slope at the point whose abscissa is x. 

Although it is logically unnecessary, it is practically helpful to pict- 
ure the differential quotient as a possible veloct¢y of a possible s/ope. 
Of the two independent discoverers of the Calculus, Newton seemed 
to have employed the former image, and Leibnitz the latter. New- 
ton’s term for a differential quotient was “ fluxion.” 


EXAMPLES, —1. Find the derivatives of x12, x°, x2, 2. What is the 
meaning of the answer to the last ? 

2. How many times as fast does y increase as x when y=! and 
wis 2'? 

3. How fast does x® increase compared with « when x is —1? 
What does the negative answer mean ? 


GENERAL METHOD OF DIFFERENTIATION 13 


18. The process employed in this chapter for obtaining 
the derivative of a function is called the “general method of 
differentiation.” It consists (1) in giving to the independent 
variable a small increment, thus causing another small incre- 
ment* in the dependent variable or function ; (2) in writing 
the relation between the two variables first without and then 
with these increments and subtracting the first from the 
second ; (3) in dividing through by the increment of the in- 


dependent variable ; (4) in passing over from ee to = 


This process should be thoroughly mastered by the 
student, for it contains, in embryo, the whole of the Infini- 
tesimal Calculus. 

He will observe that the order of steps (3) and (4) cannot 
be inverted without producing the barren result o = o. 


19. Nevertheless, we can anticipate the result of step (4) 
without changing from the form of (2). Thus, the equation 


JHS HAH +30 +52" 
yields at step (2): 
Ay=2Ax+6x« Ax+3(Ax)?+15 x°Ax+15 x(Ax)?+5 (Ax)? 
=(2+6x%+ 15 x*)Ax +(3 + 15 x) (Ax)? + 5 (Ax). 

It can readily be foreseen that step (3) (z.e. dividing by 
Ax) will remove the first Ax, and reduce the exponents of 
the powers of Ax by one, and that therefore when step (4) 
is performed (¢.e. reducing Ax to zero), all terms beyond 
the first will disappear, leaving 2+6x«+15.° as the 


derivative. Now it is clear that this result could have been 
anticipated simply by weglecting the terms involving powers 


* Decrements may always be regarded as negative increments, 


14 INFINITESIMAL CALCULUS 


of Ax higher than the first, and taking the coefficient of the 
first power as the required derivative. 

Though this process of neglecting certain terms at step 
(2) is a mere anticipation of what must necessarily happen 
at step (4), it may be shown to be perfectly natural zz s7¢u. 
If Ax be less than one, (Ax)? will be less than Ax, and 
(Ax)* less than (Ax)’, etc. By making Ax smaller and 
smaller, the higher powers (Ax)’, (Ax)*, etc., can be made 
indefinitely small, not only absolutely, but zz comparison 
with Ax. The higher powers of Ax thus growing negligible 
relatively to Ax, the terms in which those powers occur as 
factors must also grow negligible (provided, of course, the 
other factor composing each such term does not approach 
infinity as limit). 


Thus, if Ax is 745, (Ax)? is ggggq, and (Ar) only zggq55- ~Con- 
sequently in the equation 
Ay=(2+6%+15 x2)Ax+(3+15 x) (Ax)? + 5(Axr)3, 


we can, by reducing Aw sufficiently, make the terms beyond the first 

as small as we please compared with the first, no matter what be the 

value of x, so long as it is finite, thus keeping the parentheses finite. 

For instance, if « be 2, we have Ay = 74 Ax + 33(Ax)? + 5(Az)3, 
Then, if 


Ax be .o1, this becomes 
Ay = .74 + .0033 + .000,005. 
If Ax = .oo1, it becomes 
Ay = .074 + .000,033 + .000,000,005. 
If Ax = .000,001, it becomes 
Ay = .000,074 + .000,000,000,033 + .000,000,000,000,000,005, 


and the smaller we make Ax, the more negligible become ‘the terms 
involving (Av)? and (Ax), until at the limit they become, not simply 
negligible “ for practical purposes,” but adso/utely negligible. 


GENERAL METHOD OF DIFFERENTIATION  °15 


The anticipatory neglect of terms involving powers of Ax 
higher than the first often saves a great deal of labor. 


EXAMPLES. 


1. Find oy hen ph ma 
ax 


2. Find ay when y= 47+ 826+ 4, 
ax 


3. Find @ when i= wi Ocy, 
ax 


4. Find & when y= ax™ + bx", m and m being constant and 
integral. ° = Ans. amx™ 14 bnxr-l, 


5. If x, the side of a square, has an increment 7, what will be the 
increment of the area of the square ? 


6. In the function y = 3 «? + 2.4, find the value of + when y in- 
creases 20 times as fast as x. ab Dias sherciee & 
Differentiate the following functions: 
AY, = 23 GIA 1c. 
8. y=4x%°—7234 24-2, Ans. 20 x* — 21 x24 2, 
9. y=3x2-(a4 dx. 
10. y=(6 + x)3 — dx?. Ans. 30+ 4 bx + 32%. 


16 INFINITESIMAL CALCULUS 


CHAPTER II 
GENERAL THEOREMS OF DIFFERENTIATION 
20. If we differentiate 
Yao 
by the general method, we obtain 


Oa, (eo) 


Clearing this equation of fractions, we have 
ay = 2 ax. | (2) 


This last equation is simply another form of the first, and 
more convenient for some purposes, 


Thus, dy = 6 xdx is a transformation of 


ey — Orr, 
ax 

which in turn means lim Ay = 612: 
Ax 


6x isa differential quotient and 6 xdx is a differential. 

These conceptions are strictly correlative. To obtain the differen- 
tial quotient from the differential, we simply divide by dx; to obtain 
the reverse, we multiply by dx. 


GENERAL THEOREMS OF DIFFERENTIATION 17 


EXAMPLES. 


1. What is the differential of x5? 


2. The differential quotients of «7, x1, x4? 


21. To expréss the mere fact that y is a function of x, 
without specifying exactly w/a? function, it is customary to 
use the letters /, 7, ¢, w (and rarely others) followed by x 
in a parenthesis. They may be regarded simply as abbrevia- 
tions of the word “ function.” Thus 


y = Function of x 
is abbreviated to yes K(x), 


It is to be observed that the letters 7, f ¢, y, etc., do not repre- 
sent quantities like x and y, but, like A and d, represent operations 
on quantities, 


22. The general expression for a function, such as $(x), 
is often used to express, within brief compass, any sfecza/ 
function. Thus if we have the equation 


rae he ye ga aa 

na ax” 

a3 sx a-xe? 
oe 4x? 


we may shorten this to y= (x) by denoting the clumsy 
right-hand member by ¢$(). 

Again, if we have a definite curve, such as a statistical 
diagram, whose coordinates we call x and y, we may use 


y=S(x) 


to express the fact that y is related to x in the particular 
manner delineated by the curve. 


18 INFINITESIMAL CALCULUS 


23. The differential quotient, or derivative of a function 
of x, is itself a function of x. 
To denote the differential quotient of 


F(x), 
we use the expression Lil). 
Thus let @(x) stand for x®, 
Then @/(x) stands for 62°. 
The aifferential of (x) is therefore expressed by 
ENCANA 


24. Another method of expressing the differential quo- 


tient of 
F(x) 


connects it with the general method of differentiation. Thus, 
if x receives an increment Ax, /(x) will become 
FF (x + Ax). 
This differs from its original value /(x) by 
F(x + Ax)— F(x). 
The ratio of this increment of the function to the incre- 
ment Ax, of the independent variable ~, is 
F(x + Ax)— F(x) 
Ax 
F(x + Ax)— F(x) 
3 


Its limit, viz. lim 
. PN 


is the differential quotient of A(x) ; z.e. is 
YMC & 
The above process is identical with the general method of differen- 


tiation, though we have expressed it without the use of y. We might 
have proceeded as follows: 


GENERAL THEOREMS OF DIFFERENTIATION 19 


Put /(«) equal to y so that 
y= F(x). 
Subtract this from y+ Ay= F(« + Ax), 
and divide by Ax, giving 
Ay _ F(x + Ax) — F(x) 


Ax Ax 

or, at the limit 
cei eG Sina Ax)— F(x) 
dx Ax , 


25. Yet one more notation should be familiarized. 
Rather it is a new application of an old one. Instead 


of writing 2, we may replace y in this expression by 
x 


F(x), so that it reads 
a[ F(x)]. 
ax 


The student will do well now to release his mind from y as any 
necessary element in the analysis. It is to be regarded merely as a 
further abbreviation of /(x). 

/’(«) rather than y is to be thought of as primarily the function of 
(x). Thus, in our introductory example, instead of denoting space by 
_ sand writing s = 1622, we need only say if ¢ denotes time, the function 
of 4, 162%, will denote space. 

So also if x denotes the abscissa of a curve, /(x) instead of y de- 


notes its ordinate. 
ax") 


Thus, ea 
ax 
or NC cel Memo Th of kop 
d (x?) 
ae ——/=—? ile | 
EXAMPLES. oa d (x*) 


We thus have five methods of denoting the differential 
quotient of y, or its equal /’(x) ; viz.: 
Ay ad al[F(x)] me FAY) = F(a), 


> 2 , Mei aes HOLE Le > i 
tee NN ax epee Ax 


20 INFINITESIMAL CALCULUS 


26. If a function of x is the sum of several functions of 
was aan BY ; 
F(x) =f (*) +h(e) +e; 
then, since this equation holds true of all values of x, it 
holds true when x becomes «+ Ax, so that 


F(x + Ax) =f (~ + Ax) +f,(~ + Ax) + ++ 
Subtracting the upper equation from the lower, and divid- 
ing by Ax, we obtain 
F(a+Ax)—F(x)_ file+Ax)—A(s) 
Ax it Ax 


ae =i) ee 


Now let Ax approach zero as its limit. Then for the 
limits of the terms in the above equation, we have: 


_ £(xe+Ax)—F(x) . file+tAx)—A) 
Li PRA 


or EF" (x) =f (x) +fil(x) +, ete. 


That is, the differential quotient of the sum of several func- 
tions ts the sum of the differential quotients of those functions. 
The same reasoning establishes the corresponding theorem 
for the difference of functions. 


+, etc. 


Thus the differential quotient of +?-+ #3 is 2% + 3 x?. 
Sometimes the theorem is used in the differential form 


Fl! (x) dx = fi! (x) dx + fo! (x) dx + ++, 
or again F'(x)dx =[ fil(*) + fa! (4) + +++] de. 
EXAMPLES. — Find the differential quotient of: 


1. x5 + 4? — x4, 2. «'— x? +x, 3. — 227+ 412, 


~“ 


GENERAL THEOREMS OF DIFFERENTIATION |. 21 


27. Ifa function of x is the sum of another function of 
x and a constant quantity, z.e. if 


F(x) =f(x) + &, (1) 
where X is a constant, then 
F'(x) =f"); (2) 


the same result as if X were not present in (1) at all. The 
proof of (2) is simple. When x becomes x + Ax, (1) be- 
comes 


F(« + Ax) =f(x + Ax) + K, Cie 


When we subtract (1) from (1)', X disappears entirely, and 
we have, after dividing by Ax, 


F(x + Ax) — F(x) _ Se + Ax) — f(x) 
Ax Ax 


which reduces at the limit to (2). The same result would 
be obtained if in (1) A were preceded by the minus instead 
of the plus sign. 

Hence, to obtain the derivative of the sum (or difference) 
of a series of terms, some of which are constants, we simply 
take the sum (or difference) of the derivatives of all the 
terms which are functions of x, ignoring those which are 
constant. 


. dy 
= ia8 — 2x2, 
Thus, if Six Kk oae 5, 5 
Again, the derivative of 


e—a«att+xta—b—8 is 5xt—4x341. 


The foregoing result is sometimes expressed by regarding 
all the terms, even the constants, as functions of x, and 
saying that the derivative of a constant term is zero. 


22 INFINITESIMAL CALCULUS 


EXAMPLES. — Find the differential quotient of: 
1, z7+ 2, 2. 2727 +34 x4, 3. «3+ 25+ 19, 
4. Prove last by general method of differentiation. 


28. If a function of x is the product of a constant by 
another function of x, z.e. if 


LG) == Kh a), (1) 
then Py eas ore (2) 


that is, the derivative of the product of a constant by a func- 
tion ts the product of the constant by the derivative of the 
Junction. 


Proor. — When x becomes x + Ax, (1) becomes 
F(a + Ax) = Ko(x + Ax). (7)! 
Subtracting (1) from (1)' and dividing by Ax, we have 
F(x + Ax) — F(x) _XK¢ (x + Ax) — Ko (x) 


Ay Ax 
Ree + Ax) — (x) , 
Ax 
or at the limit, Pia he Gee 


Coro.LLtary. — The derivative of mx” is m times the de- 
rivative of x", as given in § 16. Hence, it is mmx*"". This 
result is so often used that it should be carefully memorized. 
When 2 is 1, the derivative is simply #. (Show this directly, 
by § 18.) 

EXAMPLES, — Differentiate 

SAO Neel, Cae a a kee 


x6 wee (14 V5 ): 


’ a 


I—V2 


GENERAL THEOREMS OF DIFFERENTIATION 23 


29. If a function of x is the product of two functions >f 
ceed n= > (x) u(x), then 
F(a + Ax) = $ (x + Ax) (x + Ax). 


Subtracting and dividing by Ax, we have 


F(a + Ax) — F(x) _ b(x + Ax) p(x + Ax) — P(x) p(x) 
Ax Ax 


The right member may be changed in form without suf- 
fering any change in value by adding and _ subtracting 
b(x)y(« + Ax) in its numerator, giving 


o(xt+Ax)y(x+Ax)— (x) V(x) —b(x)W(x+ Ax) +O(x)¥ (a+ Ax) 
Ax 


Grouping the terms according to common factors, we 
have 
[p(x + Ax) — or) ]¥@ +Ax) + 6()Lb or +42) — ¥@)] 
Ax ; 
or 


Lo (x + Ax) — $(x)] 
Ax 


yeaa) + ETO) HH) 50, 


Taking these terms in order, we see that the 


limit of A ae es, $'(x), 
limit of W(x + Ax) is y(x), 


limit of dale eta is y'(x), 


limit of (x) is d(x), 
which gives for the limit of the right member of the equation 


b' (x) W(x) +(x) o(x) 5 


24 INFINITESIMAL CALCULUS 


while for the other (or left) member of the equation the 


F(x + Ax) — F(x) 


ce is £'(x). 


limit of 


Putting these limits equal, we have 
B(x) = $x) W (a) + Y'(%) o @). 


In words, the derivative of the product of two functions ts 
the sum of the products obtained by multiplying the derivative 
of each function by the other function. 


thus,  C2C+ 291 dO) G4 42) 4 Mt) 0 
ax ax 7 


= 2x(1+ 27)4+ 2% - x? 
2 a(t 2) 


ExAmpLes.—1. Find the derivative of (1 + x?)(1 — x?) first by 
§ 29 and afterwards by multiplying out and then differentiating. 


2. (2 + x3 —a4)(5 425), 422 +1) (49-2), 
a(342-+4)(548 + 622+ 74+ 8), (a+ 6)(kx™+ ha™+ p) (gx? + 7). 

3. Prove § 28 by using § 29, regarding & as a form of ~(x), whose 
derivative is zero. (See § 27, end.) 


4. Prove § 29, using a different notation, 


30. CoroLiary. —If F(x) = fi(*) fo(*) (4%), we may abbrevi- 
ate fo(7) /g(x) to p(x), so that 
| F=f) $(2), 
whence f(a) =f! (4) Oe) + Oe) AC: 
Replacing ¢(x) by its value 72(4) fg() and ¢!(x) by its value 
fal (x) faa) + fs! (4) A), 


we have 
Pae)=/'@)[AMADI+ LAMA +A'MAMIAC 
=A'OAMAM +A'OAMOAG) +A' MAGA). 


GENERAL THEOREMS OF DIFFERENTIATION 25 


By successive applications of § 29 this theorem can be generalized 
to the product of any number of functions, and in words is as follows: 

The derivative of the product of any number of functions is the 
sum of the products obtained by multiplying the derivative of each 
function by the product of all the other functions. 


EXAMPLES. — Find the derivatives of 
(2+ 1)(e+1)(e—1), 28424 2« 4+ 3)(244— 7) (4 — 4°). 


Sle lt (7) = GY and (x) is not zero, then 
I I 
P(x + Ax)— F(x) o(*+ 4x) $(%) 
Ax oh Ax 


_ $(%)— o(* + Ax) 
~ Ax o(x) h(x +Ax) 
a et! o(x+Ax) —$(x) 
Meee Ax 


$'(x) 


or at the limit 4’(*+)= ———. 


[se oF 
Bers (oo) a 
~ [6(a) 
That is, the derivative of the reciprocal of a function is 
minus the derivative of the function divided by the square 
of the function. 
Thus the differential quotient of “4S is 
3x 


— 4(3 x7) 
ax SORE 6x 4 = Gales 
(347)? WE ‘3a 


32. EXAMPLES. 
1. Find the derivative of 
I I I 4 Ee Bt Nal I+24 


, ——, ———..__ Ans. : ESA A 
aA 14+ 14-2442 x aren + 43)? (1 +x + x2)2 


26 _  LINFINITESIMAL CALCULUS 


2. Show by method of § 29, that if 


Pele. as Oa): 
Sere 
then Fi lh Acct a 
(Y)? 


where the (r)’s are omitted for brevity. 


3. Prove the same theorem by applying results of §§ 29, 31, after 
p 


throwing 7 in the form ¢ -—- 


33- We may interject here an application of the result of § 31 
to generalizing the theorem of § 16. The differential quotient of +” 
was there obtained only under the restriction that 2 be a positive 


: ‘ iia I ‘ 

integer. But if 7 be a negative integer, —7, then 2” becomes —, This 
am 

fraction has meaning only provided the denominator is not zero, z.e. x 


is not zero. The differential quotient becomes 


— mxm-1 


arm = 


which reduces to — mx—™-! or nxr—l, 


That is, the restriction imposed in § 16 that 2 must be 
positive, may be removed. 


EXAMPLES. 
1. Differentiate x-?. 2. Differentiate 3x75. 
3. Differentiate +. 4. Differentiate —1. 
a 8x3 
34. If we wish to differentiate the quotient of two func- 
tions as Gk we can do this by combining the results of 


W(x) 
$$ 29 and 31, for the quotient may be written (2) - me y 
a 
1+ x2 


‘ is obtained by writing it 
I—.x 


(1 + x?) a Applying the theorem for products, we get 
I—<x 


Thus, the differential quotient of 


Pe (pith) 
reagea) I d (1 + x?) 
ax ey =] dE 


which can readily be reduced. 


(+23) 


GENERAL THEOREMS OF DIFFERENTIATION 21 


If the student prefers, he may simply memorize the result of example 
2, § 32, and apply. 

35. Ifzisa function of y, and y of x, an increment Ax 
of x produces Ay of y, which in turn produces Az of z. 
ie Ne Ay 
Ax Ay Ax 

The limits of these magnitudes (assuming that definite 
limits exist) will therefore have the same relation, viz. : 


Evidently 


We a2) by. 
dx dy ax 
This may also be expressed : 
If Mx) = o[ /(x)];, 
then F(x) = $'[/(@) J"): 


It must be carefully noted that $/[ f(x) ] means the derivative of 


| /(«) ], zo¢ with respect to x, but with respect to f(x). It is ® vot 
dol say], HOLS), 2 
df (x) dx i 
In words, the derivative wth respect to x of a function of 
a function of x, is the derivative of the former function zw7¢h 
respect to the latter, multiplied by the derivative of the latter 
with respect to x. 


Z AY EV 
—, or again it Is 
ax’ 8 


Uist yaa (1 x2) 2, e4 may be found by denoting (1 + «?) by w, 


and then finding &. from y = w*, and we from w=1-+4 2x7. Whence 
a 


Ww 
es = = Bt 2 = 31 + x7 )22 x. 


But the use of w is quite unnecessary, and the student should learn 
to dispense with it as well as with y also. The required derivative then 
a(1+.2«7)3 a(1+ 2x?) 
d(1 + x) ; ax 

Employing the notation of differentials, the process is even 
more easily remembered and applied. ‘The differential of 


oL/(*) ] is 
dp| f(x) ] or $'L f(x) a(x), or CHa) 1 (aye. 


Pa Gle ee) 2x. 


becomes 


28. INFINITESIMAL CALCULUS 


That is, we first differentiate, treating “/(x)” as a single 
character, and our result contains d(x). We then perform 
the further differentiation indicated by this d/(x). 

Thus, a(t + x?)3 = 3(1 4+ x?)2d(1 4 x?) 

= 3( 1 Px oa, 
where “(1 + x?)” is first kept intact as if it were not a combination of 
symbols, but a single cumbrous symbol. 

36. Examples. 

1. Differentiate 4(2 + +*)%, 

Differentiate (7 + x)°. 

Differentiate 2(1 + 24+ .47)8, Ams. 1201+ 4)(1 + 2x44 27). 
Differentiate (3 +3 — 2)-4, 

eer ANE 

(0? + +1)? 


Seats Basse 


Differentiate 
en le Beatin LP ANS = _9044+T) 
(2*3 4+ 347+ 4)° (24% + 327+ 4)° 
7. Differentiate a+ 6(1 + «?)?+ c(1 + #7)§ + A(1 + x7), 

8. Differentiate 


6. Differentiate 


(3(ax?+ dx +6)? + )b—m(axt-+ bx +o"). 


5 
k(ax? + bx +c)? 


37. In like manner, if we have a function of a function of a func- 


tion, as Ka) =o Aa), 
we may show that F(x) ='[&(«) él (x). 


Substituting for € its given value and for é! its value as obtained by 
§ 35, we have 
F(x) = AOD TAS), 
and so on for any number of functions. If we use differentials instead 
of differential quotients, we have 


A b1(G2[ 8 (+++) ])} = o1'dhe 
= $1'$2'dbs 
= $1'p2'bs'db4 
= etc. 
The proof is left to the student. 


GENERAL THEOREMS OF DIFFERENTIATION 29 


EXAMPLES. 
1. Find the derivative of 

4{2(1 + x7)? + 3(1 + 4?)8 4 5 fa(r + x7)? + 3(1 + x?) 8, 
2. Differentiate {a +[6+(¢ + hx?)3]>, 


38. The results of this chapter may be thus summarized : 


p SMG) EIS) £ dH) ENG) £ 


‘ cise) $(x)y'(«) + We) o'(x). 


3, AAS) _ Kg1( 


al 5, _=9@), 
de To} 


5. LEP) oA) 


30 INFINITESIMAL CALCULUS 


CHAPTER Tit 
DIFFERENTIATION OF THE ELEMENTARY FUNCTIONS 


39. We have learned (§§ 16, 33) that the derivative of 
x" is nx", where m is any integer. 2x” is the elementary 
algebraic function. 

We have now to differentiate elementary functions called 
“{ranscendental.” To do this we recur to the general 
method of differentiation. We first take up the trigono- 
metric functions. 


40. q(si oe Gin sin (x + Ax) — sinx 
ax Ax 
— Jim S10 cos Ax + cos x sin Ax — sin « 
Ax 
Ba { sin Ax ‘ I eet) 
= lim < cos x ——— — sin x « ————— 
x Ax 
B sin Ax . ae 
ut ———— becomes unity at the limit when Ax becomes zero, and 


I — cosAx 
-_——___ pecomes zero. 
BX 


These are shown by means of Fig. 4, where 4B is an arc Ax on a 
unit radius OA. So that AC is sin Ax, CO is cosAx, and CA is 


I — cos Ax. 
BC 


sin Ax 
aS J 


AX 


is therefore 


I —cosAxr is CA 
Ax BA 


and 


‘ DIFFERENTIATION OF FUNCTIONS ot 


When 4A becomes zero, CA and BC become zero. The proof 


that lim = —— NP rb te ise a) oA = 0, is left to the student with the follow- 


ing hints: 


Fic. 4. 


BC Se CO 
arc b4 8 7DA V4A0 


2. Be sig cal FEES BC which approaches 0 X I. 


BA RB CteBAGCE BA 


—, which approaches I as limit. 


d(sin x 4 
Hence ASD = cos.x x I —sinxxo 
Ix 


= COS &. 


In like manner, we may prove 


da cos x 
ax 


a( sin x 
41 dtan x = cos x 
Me 2a we. Cia 


__ cos x cos x + sinx sin x 
cos? x 


— sin x. 


eee: 
cos? 7 
d(cotx) — 


Similarly, ss core 


32 INFINITESIMAL CALCULUS 


I 
yee 
GSC fs Ge 


Die = i 
4 pe oe etc., according to § 31, 
pion 
F a(cosecx) — \sin x} 5 : 
an ae PMT TIET er 
d(a*) Z qz¢~t+Ar _ gt 
2 Y= lim ——— 
43 ax Ax 
‘ a | 
= lim a? . ——__—_- 
jhe 
Now let asx — 1 = 64, so that a4*=14 6, 
and Ax log a = log (1+ 4), 
and pi OR LEI OY, 
loga 
Da) ine 5 
Then YET lim a log(1 +0) 
loga 
= ]j x} ; 
= dima Raper apres 
) 
: I 
= lim’ e* lop @— 


log {(1 + 8)*} 
1 
The limit of (1 + 6)4, when 6 becomes zero (which evidently occurs 
when Ax becomes zero) is 2.718 approximately, and is called e.* 


* This fundamental magnitude may be-pictured as follows: Suppose 
interest is at 4% corresponding to “25 years purchase.” $1 com- 
pounded yearly for this 25 years amounts to (1.04)25, Compounded 
half-yearly for the same 25 years, it is 02 a erie (1.01) 


daily (i + Fe OR ei 5-7 momently, lim (1 + aS; or é. Thus é is 
simply the amount of $1 at momendly or continuous interest during the 
“purchase period.” This is $2.718, whereas with gwarter/y com- 
pounding the amount would be $ 2.705, and with yearly, $ 2.666. 


DIFFERENTIATION OF FUNCTIONS 33 


Hence at the limit, a a*) =a*loga PEAS, 
ax loge 


This result is independent of the system of logarithms. It is true 
of “common logarithms.” If we take e as the base (7.e. employ the 
Naperian system), then log e = I, and the result simplifies to 


dhe a log a. 


ax 
Finally, if a = e, the result is still simpler, for loge=1. We then 
have é 
GS) ) —e'5 
ax 


Henceforth we shall denote common logarithms by 
“ Log’’ and Naperian logarithms by “log.” Any other sort 
of logarithms will be denoted by “ log,,”” where the subscript 
é denotes the base of the system. 


44. We now proceed to the inverse functions of those just considered, 
y = arc sin x, means that y is the arc whose sine is « (sometimes 
the notation sin=! x is used), z.e. it means the same thing as 


wee Sill 4, 
From this a cos y 
ay 
=I — sin? y 
=V1 — x4, 


But BOTs) the reciprocal of ve, since these expressions are the 
dy ax 


x 


limiting values of Beyond oe which are reciprocals. 
Ay Ax 


Hence a : 
dx Vi 
Or @(arc sin x) _ Tay vey 
dx Vi — x 
Similarly, d(arccosx)__ I 


dx Wiens 


34 INFINITESIMAL CALCULUS 


45. Ify=arc tan z, then # = tan y. 


ax I 
dy  cos*y 
= sec*y 
= 1+ tan*y 
=1+ 242 
Hence Cae 
rh nat Ave OE 
On d(arctany)_ ot 
ax 1+ + 
Similarly, d(arc cot x) ear te 
ax 1+ x2 


46. Ify=log.«x, then x = ZY, where 4 is the base of the system. 


Hence Bas log,d oe 
dy loge 
But 109; 2.71. 
Hence Ba _ by — 
dy logye 
= = . 
logye 
Hence dy _ logee 
ax x 


This is independent of the particular system of logarithms. 
If =e, then log,e = 1, and the result simplifies to 


7. x 


ax a 


47- We may now still further generalize the theorem expressed in 
§§ 16, 33. The number z has been restricted to an integer. But if 
y = x" where z is any real numeer, 


a e 


then log y = # log x. 
Taking the differential of each side, 
dy ax 
—=—. == H ——e 
J ¥ 


DIFFERENTIATION OF FUNCTIONS 30 


Hence cere eats 
Phone He 


WX 


x 
= nxn-l, 
That is, the restriction of §§ 16, 33, that # must be an 


integer is now removed. It may be a fraction, an irrational 


number, or any real number whatever. 


EXAMPLES. 
1. What is the differential quotient of 
1 


3 Hh ph oe 5 ily. ys BD 
Dee fie SCA Vx, Vx, x Aas 


3 
Oe ORs Viriat, (et 91)8, Vache Ve cx? 


48. The results of this chapter may be thus summarized : 


DIRECT FUNCTIONS. INVERSE FUNCTIONS, 


TOA E ES 
AOL ol meet (Ea ig 


a(sin x)= cos xdx. d(arc sin x)= ee ‘ 
I—x 
: — ax 
d(cos x)= — sin xdx. d(arc cos x) = = 
I— x 
ax ax 
at ——_- d(arc tan x)= . 
aus) mea ( ) ee 3 
a(cot x)= = d (arc cot x)= =a 
sin? x 1+x 
Ale a’ Log adx 
(a*) = ——2—— a (Log x)= “Loge 
Loge 
ax 
= a’ log adx. d(log x)= = 


Me\ =e dx 


36 INFINITESIMAL CALCULUS 


No function inverse to x” (or to the more general form 
kx”) is given, since in this case the inverse is identical in 
form with the direct function. 

1 
(Thus, if y= 2", x = 7" = y™, a form identical with 2", its inverse.) 
49. EXAMPLES. 


1. Differentiate 3 sin x. 


2. Differentiate 1 — asinx + Jdcos-x. 

3. Differentiate 2sinxcosx. Ans. 2cos2x. 

4. Differentiate sin x tan x. 

5. Differentiate cot « + x? cos x. 

6. Differentiate log x +tanxcosx. Ans. T+ cos x. 

7. Differentiate «72%. ; 

8. Differentiate (2 log x — dx? + ca™)(1 — x). 

9. Differentiate sin34. Ams. 300s 3x. 
10. Differentiate cos x?. 
11. Differentiate tan (1 + x++ 2x7). Ans. fee eee 

cos?(1 + x + x*) 

12. Differentiate log #3 + +x tan («+ a® — arccos 34). 


SUCCESSIVE DIFFERENTIATION 37 


CHAPTER IV 
SUCCESSIVE DIFFERENTIATION — MAXIMA AND MINIMA 


50. The derivative of 2x* is, as we know, 8x*. The 
derivative of 8 x* is, in turn, 24x. The derivative of a 
derivative is called the second derivative of the original 
function. 

When /(x) stands for the original function, and /'(x) 
for its derivative (to avoid misunderstanding we must now 
call it the frst derivative), then /(x) denotes the second 
derivative, and /''(x) the third derivative (z.e. the deriva- 
tive of #''(x), etc. 


Again, if we use the notation a for the first derivative, 
ax 


AG) 
*/, which is usually 


ey (Ly 
ax? 
ax 


2 
abbreviated to eh likewise the third or 


the second derivative is evidently 


is written 


3 4 5 
gO. and so on to Ey, ee etc. 
"bee 1h olay fo 


51. EXAMPLES. 
1. What is the third derivative of x? 
2. What are the 2d, 3d, 4th derivatives of «?? 


3. Differentiate successively ax". When, if ever, will the answers 
become zero? What sort of a number must 7 be to bring about such 
a result? 


4. Differentiate successively sinx. Ans.cos x, —sin x, —cos x, sin.x, 


38 INFINITESIMAL CALCULUS 


Differentiate successively tan x. 
Differentiate successively @*- 


Differentiate successively arc sin x. 


OI AH SH 


Differentiate successively arc tan x. 
ae oe 
Ans. : hase ee) epee 2( tee 
I+ x (1 + x7)? (1 + x?) 
9. Differentiate successively log x. 


52. Just as the first derivative threw light on the problems 
of velocity, tangential slope, etc., so the second derivative 
will illuminate acceleration, curvature, etc. 

We have seen that if for a falling body s = 16 7”, then 


as 


“~=32¢ 
at eed () 
a’s 
hence eae 2 
whenc pet (2) 


We may understand this result better if we designate e 


by v, as in § 6, so that (1) becomes 


rf WH (1 
and (2) lee a2: : (2)! 
| at 
where “ is evidently simply 
at 


for both are mere abbreviations of 


a(& 
at) 


at 


What does equation (2) or (2)' mean? @ means the rate 


at which the body ts gaining speed. It is clear that moving 


a 


SUCCESSIVE DIFFERENTIATION ae 


bodies do gain or lose speed, and that some gain or lose 
faster than others. - 

The gain or loss of speed has nothing to do with how fast 
a body is going. A slowly moving body may be gaining 
speed very fast, while a fast moving body may not be gain- 
ing at all, or may even be losing speed. | 

If we use the term veo to indicate a unit of velocity, or 
one foot per second, we know from (1) that a body which 
has fallen 2 seconds has then a speed of 64 velos, while at 
the end of 5 seconds its speed is 160 velos. Here is a gain 
of 96 velos in 3 seconds, or an average of 32 velos per 
second. 

This does not, of course, imply that the body had gained 
at the rate of 32 velos per second all the time. But equa- 
tion (2) tells us that this is the case. A falling body on the 
earth is constantly gaining velocity at the rate of 32 velos 
per second. 

Rate of gain of velocity is called acceleration, and we see, 
therefore, that a falling body is a case of “ uniformly accel- 
erated motion.” 


Observe that the acceleration or rate of gain of velocity expressed in 
32 velos per second, cannot be expressed as any number of feet per 
second. On the contrary, substituting for the word “velos” its defi- 
nition “ feet-per-second,” we see that 32 velos per second is 32 feet per 
second per second. 

If the distance a body moves in time ¢ is not 162%, but 107%, then its 
velocity is 302”, and acceleration 60¢. In other words, its acceleration 
in this case depends on the time. If the body has fallen 2 seconds, its 
acceleration is 120 velos per second; if 3, 180 velos per second; etc. 


53- If /(«) expresses the ordinate of any point on a 
curve when the abscissa is x, we have seen that /''(x) 
expresses the tangential slope at that point. What does 
F''(x) represent? Evidently the rate at which that slope is 


40 INFINITESIMAL CALCULUS 


changing at that particular point as x increases. It denotes 
what we may call the curvature at that point with respect to 
the axis of x. 


A A 


A 
Fic. 5.— A, rate of gain of slope positive ; B (“‘ point of inflection”), zero ; 
C, negative. 

Curvature, however, is usually measured with respect to 
the tangent itself. The expression for this, the more proper 
sense of curvature, is somewhat more complicated. At a 
point when the curve is horizontal, the two sorts of curva- 
ture are identical. 


54. When the curve is horizontal, the slope of the tan- 
gent /''(x) is, as has been seen, zero. But the curve may 
be horizontal at three sorts of points: a maximum as at 4 


A 


c 


Fic. 6. — Points of zero slope: A, maximum; B, horizontal point of inflection; 
C, minimum; YD, maximum. 


and D (Fig. 6), or a minimum as at C, or a horizontal point 
of inflection as at B. 

A maximum point on a curve is a point such that the 
ordinate, or y, of that point is larger than the ordinates of 
points in its neighborhood on either side. (The phrase 


SUCCESSIVE DIFFERENTIATION 41 


“ points in its neighborhood” means all points on the curve 
within some small but finite distance on either side.) A 
minimum point is one whose ordinate is less than the 
ordinates in its neighborhood on either side. A point of 
inflection is one where the neighboring parts of the curve 
on opposite sides of the point are also on opposite sides of 
the tangent as at 4 in Figs. 5 and 6. 

In the neighborhood at the left of a maximum the slope 
of the curve is positive, while on the right it is negative. 
For a minimum, the slope is negative on the left and positive 
on the right. For a horizontal point of inflection, the slope 
is positive on both sides or else negative on both sides. 

It is to be observed that a curve may have more than one maximum 
or minimum, and that a maximum ordinate does zzof mean the greatest 


ordinate of all, but only the greatest 2 7zts nezghborhood. Thus the 
ordinate at D is a maximum, though that at 4 is larger. 


55- Dropping the symbolism of the curve, it is clear that 
when a function (x) reaches a maximum or minimum, then 
F'(x)= 0, for ''(x) represents the rate of increase of F(x), 
and at a maximum or minimum this rate is zero. 

But if, conversely, we have /'(x)=0, we simply know 
that for that particular value of x which satisfies this equa- 
tion /(«) is not increasing nor decreasing. We cannot tell 
whether it is a maximum or a minimum or an “ inflectional 
stationary” value (¢.e. one such that /’(x) will increase for a 
change of x in one direction and decrease for a change of x 
in the other direction). 


56. Now these questions can be settled by recourse to 
the second derivative, provided this is not also zero. 

If the second derivative be positive, the function is a 
minimum ; if it be negative, itis a maximum. This will be 


42, INFINITESIMAL CALCULUS 


clear if we remind ourselves of the meaning of the second 
derivative. It indicates the rate of change of the slope. If 
positive, it means the slope is increasing; if negative, it 
means the slope is decreasing. 

If, therefore, at a point where the first derivative or slope 
is zero, the second derivative or “curvature” (§ 53) is posi- 
tive, we know that at that point the slope is zmcreasing. But 
as its present value is zero, it must be changing from a nega- 
tive to a positive value. ‘This can evidently only occur at a 
minimum. /¢r contra, if the second derivative is negative, 
it indicates a slope growing /ss, z.e. (as the slope is now 
zero) changing from positive to negative. This evidently 
occurs at the maximum, and nowhere else. 


Thus, take the function +? — 27%. This has for first derivative 
3 x* — 27, and for second derivative 6x. Putting the first expression 
equal to zero and solving, we find x =+ 3; that is, the function 
x? — 27x has two points at which it is stationary (or the tangent is 
horizontal), where x is 3, and where x is — 3. The first of these is a 
minimum, and the second a maximum; for the second derivative 6 x is 
positive for x = 3, and negative for x =— 3. 


57. The exceptional case mentioned in § 56 (viz. where 
the value of «x, which renders the first derivative zero, also 
renders the second derivative zero) seldom occurs in 
practice. When it does occur, we cannot decide the nature 
of the function for that point, without recourse to the third 
derivative. If this be positive, the function is neither at a 
maximum nor minimum, but at a hori- 
zontal point of inflection, as at 4 (Fig. 
7), when, for an increase of x, the 

Fic. 7. function was increasing, both before 
and after the point. If, on the other hand, it be negative, 
the function is at a horizontal point of inflection as at B 


A 


SUCCESSIVE DIFFERENTIATION 43 


(Fig. 6), when the function was decreasing both before and 
after reaching this point. TIf, finally, it be zero, we are again 
left in the dark as to the nature of the function, and must 
proceed to the fourth derivative. We employ this just as if 
it were the second. If it turns out zero, and forces us to 
consider the fifth, we employ this just as if it were the third, 
and so on. 

That is, as dong as the successive derivatives turn out zero, 
we go on until we find one which is not zero. Tf this deriva- 
tive be of an EVEN order (7.e. 2d, 4th, 6th, etc., derivative), 
we know that the function is either a maximum or a mint- 
mum, and is the one or the other according as the derivative 
in question ts negative or positive. But if the derivative 
which does not vanish is of an odd order (2.2. 3d, 5th, etc.), 
we know that the function is neither at a maximum or. mini- 
mum value, but at a point of horizontal inflection and is 
increasing or decreasing according as the derivative is posi- 
tive or negative. 


58. We shall not devote the requisite space here to proving the 
truth of the last section in full, but shall merely indicate the first step, 
leaving the student, if he so desires, to extend the demonstration. 

Suppose in testing the function A(x) we find for the value of x 
which renders /’(x)= 0, that /!/!(x) is also zero, but #/!!(x) is posi- 
tive. Denoting this value of + by x1, we may state the problem as 
follows: given 


F(x) ma SD; 
Fl (x1) 20, 
JAS ye O, 


to discover the nature of (2%). 


We shall solve this by reasoning from //’! successively back to /'!’, 
F', and F. 

Since F!!'(x) is positive, it shows that F!/(x) is increasing as x 
-Increases. But as /’!/(x 1) is zero, the fact that #//(x) is increasing 


44 INFINITESIMAL CALCULUS 
shows that it was negative before reaching /'!/!(x,) and positive after. 
This is our conclusion for /". ji 

Since /'''(x) was negative before reaching /!/(x,) it shows that 
f''(x) was then decreasing, and since /’!'(x) was positive afterward, 
F'(x) was dhen increasing. 

But, if #’(x#) is zero at /'(x,) and was decreasing before and in- 
creasing after, it must have been positive both before and after. This 
is our conclusion for #’. Since /! is positive both before and after, 
it shows that /(#) was increasing both before and after, and is there- 
fore not a maximum, but a horizontal point of inflection. 


Thus let / (+) be 
x§— 6224+ 8x4 7. 


Then F'lis 4x2 —12x%4+ 8. 
Then F"' is 12 x? — 12. 
Then fling 24x, 


The roots of #!’ =o are I and—2. For x =1, F'! vanishes, but 
F''' is positive. Hence we know that / or +t — 622+ 8x47 is ata 
stationary inflectional value increasing on either side, as x increases. 

But for x =— 2, #'' is positive. Hence for this value of x, F is a 
minimum. 


59. EXAMPLES. —1. Find maximum or minimum value of x”. 
Find maximum or minimum value of 3 «? — 27. 

Find maximum or minimum value of 2427+ *«+ 1. 

Find maximum or minimum value of «7 — 127+ 6. 

Find maximum or minimum value of 243+ 647+ 6%+4 5. 
Find maximum or minimum value of «3? — 24 + 347 — 4. 

What is the nature of #* — 24 12 + 64% + 10 for = 2? 

What is the nature of #'+ 443 + 647+42%+417 forx =—1? 


Se RE TEL RAR gt 


60. If A(x) is of the form $(+) +A, where X is any 
constant, then the same values of x render /(x) a maxi- 
mum or minimum as render ¢() a maximum or minimum 
respectively. 


SUCCESSIVE DIFFERENTIATION 45 


For the nature of (x) or of ¢(x) as to maxima and minima de- 
pends exclusively on the nature of their derivatives, and the derivatives 
of these two functions (viz., @(4) + A and @(x)) are evidently identical. 

Thus to find the value of x to render 


e+ a(t | 


a maximum or minimum, we may drop the constant term and simply 
inquire for what value of x the form x? is a maximum or minimum, 


61. If /(x) is of the form A(x) when X is a positive 
constant, then the values of x which render /(x) a maxi- 
mum or minimum are the same as those which render $(x) 
a maximum or minimum respectively. 

If #(x)= K(x) where X is a negative constant, then 
the values of x which render /(~) a maximum or minimum 
are the same as those which render ¢(%) a minimum or 
maximum respectively. 


For the successive derivatives of these two functions (viz., K¢(+) 
and $(x)) are 


Ko'(x) | o (x), 
Ko"'(x) > and j (x), 
etc; CLES; 


and evidently the very same values of x will make the two first deriva- 
tives zero, and, if A be positive, will make the two second derivatives 
of the same sign or both zero; but if X be negative, will make them of 
the opposite sign or both zero. Similarly for the two third derivatives, 
etc. Since the natures of “and of ¢, as respects maxima and minima, 
depend exclusively on the signs (+, —, or ©) of their derivatives, 
the theorem is proved. 
Thus, to obtain the value of x which will make 


(: Bee ae 


a maximum or minimum, we drop the constant factor (which is evi- 
dently positive) and find out which values of « make +? — x, a maxi- 
mum or minimum. 


46 INFINITESIMAL CALCULUS 


EXAMPLES, —1. Interpret the theorems of §$ 60, 61 geometrically. 


2. Find maximum or minimum of 5(1 + x + «?)+ Io. 


17 


3. Find maximum or minimum of — 3x(x +1+ x2), 
x 


a(x? + bx+c)+e pave: 
Se ee 


4. Find maximum or minimum of m4 

62. The subject of maxima and minima is one of the 
most important in the Calculus, and has innumerable appli- 
cations in Geometry, Physics, and Economics. 


Let AAC (Fig. 8) be any triangle, and EFXZ a rectangle in- 
scribed within it. This inscribed rectangle will vary in size according 
to its position. If too low and flat, it is small. If too high and thin, 
it is also small. Between these positions there must be a position of 
maximum, where the area is the largest possible. 

Now its area is the product of the base YX or EF by the altitude 
DM, and the problem consists in discovering where 2#- DM is a 
maximum. 

To do this, we must first express Z/ and DJ in terms of some one 
variable. Out of the many possible (2.2. BH, BK, AE, FC, EH, HK, 
etc.) we select 4, and denote it by x. We call dD=4 and BC=a. 
Evidently 4D =h—«x. To express ZF in terms of x, we proceed as 
follows: The triangles 44 / and ABC are similar, so that their bases 
and altitudes are proportional. That is, 


AM _EF ,, ¢ EF 
AL ae. C h a 


SUCCESSIVE DIFFERENTIATION 47 


whence EF=“%. 
h 
Consequently EFX DM =(h— <x) i 
Z 


We wish to know for what value of x this expression is a maximum. 


We may omit the positive constant factor = leaving 
2 


(h — x)x or hx — x, 
the first differential of which is # — 2, 
which, put equal to zero and solved, gives 
i h 


PY Sis | 
2 


the required answer. 

We are sure it is a maximum and not a minimum or stationary in- 
flectional value, since the second differential is — 2; 7z.e. negative. 

We have learned, therefore, that the maximum rectangle inscribed 
in a triangle is that whose altitude is half the altitude of the triangle. 


In physics many important principles depend upon max- 
imaand minima. ‘Thus the equilibrium of a pool of water, a 
pendulum, a rocking chair, or a suspension bridge, is deter- 
mined by the condition that the centre of gravity in each 
case shall be at the lowest possible point. 

In economics we have the principle of maximum con- 
sumer’s rent, of maximum profit under a monopoly, etc. 


63 » EXAMPLES. 


1. How must a given straight line be divided so that the product 
of its two parts shall be a maximum ? 


2. What is the minimum amount of tin necessary to make a cylin- 
drical vessel which will have a given capacity 4? What must be the 
relation between the height % and the radius of the base 7? 


3. Find the maximum cylinder inscribed in a circular cone of 
revolution. Azs. Altitude of cylinder equals one third that of the 
cone. 


48 INFINITESIMAL CALCULUS 


4. Find the maximum rectangle inscribed in a semicircle. 


Ans. The sides are 7 V2, and rv 2, 
2 


5. A cylinder of revolution has a given diameter. What altitude 
must it have in order that it may have the least total area in propor- 
tion to its volume? 


H1nT. — Express volume and total area in terms of the variable alti- 
tude x, and the constant radius 7 Then find when 


total area . Cap 
——_—— 16 4 Mminimuni. 


volume 
6. If the function A/( Z) is continuous, what equation gives a value 
of # which makes the function a maximum? 
Write the algebraic expression denoting the condition under which 
the value of Z, in the equation asked for, corresponds to a maximum or 
minimum. 


7%. Ifthe price, A, of an article is fixed and the cost of producing it, 
for a given individual, is a function /(x), of the quantity produced, 
x, how much must he produce to make his profit, xf — F(*), a maxi- 
mum or minimum? Express this result in words. What condition 
must /(«) satisfy that the profit may be a maximum and not a mini- 
mum? Express this condition in words. 


8. Four equal squares with side x are removed from the corners 
of a square piece of cardboard with side ¢ and the sides are turned up 
so as to form an open square box. If the square box is to be of maxi- 


mum volume, what will be the value of x in terms of ¢2 Avs. - 

9. The distance between two points, # and C, on a coast is 5 miles. 
A person in a boat is 3 miles distant from 8, his nearest shore point. 
Supposing he can walk 5 miles an hour and can row 4 miles an hour, 
what distance from C should he land in order to reach C in the 
shortest possible time? Azs. I mile. 

10. Given /, the slant height of a right cone; find the altitude when 


the volume is a maximum. Ams. a, a 
3 


TAYLOR'S THEOREM 49 


GHAPTE RG Vi 
TAYLOR’S THEOREM 


64. We know that certain functions can be developed in 
terms of powers of variables. Thus (@+.)* becomes by 
the binomial theorem 

at+aax+6ax?+4ax°+ x. 

Again, by simple division, we may show that (provided x 
lies between — 1 and +1) 

I 
I1+.x 

Now the Calculus supplies a much simpler and more gen- 
eral method than algebra of developing functions in series 
of this sort. 

Thus, let #(«) be any function of x developable in the 
Jorm 

o(x)= 4+ Bx —a)4+ Cix—ay+ D(x —aytes, 
where a, A, B, C, etc., are constants, and the series con- 
verges. We shall show how to express the “ undetermined 
coefficients ” 4A, B, C, etc., in terms of the single constant a. 

By successive differentiation, we have * 


¢'(x) = B+2C(x—a)+3D(«*—ayrt+-: 
ani G2) ee +2-3D(x—a)+- 


etc. 


Saye gs eee ee 


* By § 26 which can readily be extended so as to apply to an infinite 
number of terms if, as is here assumed, the sum of these terms con- 
verges, 


50 INFINITESIMAL CALCULUS 


Since these equations (and the original from which they 
are derived) are true for any value of x, they are true when 


v= a. 
They then become, 
$(a)= A, Or A hid) = 
ad jeans, end oe 
a 
CR ah bee AoE ope th: 


my a 
plik Va Ter2 eg te, p-£ 4; 
etc; 


where 2! means 1-2 and 3! means 1-2-3, etc. 


Substituting these values of 4, B, C, D, etc., we have 
es 
b(2)= $(a) + $'(@) (x — 2) + 6"(@) SO 
ih ee 
+ $!"(a) ao Apc 


’ 


65. This series, which is ‘‘Taylor’s theorem,” expresses 
the magnitude of the function ¢@ for any value of x in terms 
of its magnitude and that of its derivatives for any other 
value of x. 


Thus if we could write down some exact formula y = ¢ (x) for the 
population (y) of the United States in reference to the time (+) 
elapsed since, say 1800, Taylor’s Theorem tells us that we could get 
the population in 1900, ¢ (x), merely from data of the census of 1890. 

As a first approximation we take the population of 1890 itself, ¢ (a). 
But, as the population has not remained stationary, we add a correction 
for the increase within the decade. 

This increase we first assume to be (4 — a) $!(a), z.e. the rate of 
increase known to exist in 1890, ¢/(a), multiplied by the time between 
the two censuses (* — a). Lut since the rate of increase (by which is 


ean 


TAYLOR’S THEOREM au 


here meant so many thousand souls per year, not the percentage rate) 
gla) (x — @)? 
3 
1-2 
constructed on the supposition that the rate of increase of the rate of 


increase of population, ¢/'(@), known to exist in 1890 has remained 
constant until 1900. Not content with this, we take into account the 
rate of increase of the rate of increase of the rate of increase of popu- 
lation, and so on. 


has not remained stationary, we add another correction 


66. Geometrically, the theorem states that the ordinate 
of any point of the curve y= ¢(«) can be obtained from 
the ordinate, slope, ‘‘ curvature,” etc., of any other point. 


O A B 
Fic. 9. 


Thus, OB (Fig. 9) is x and BD, ¢(x); OA is a and AC, $(a). 
The theorem tells us that the ordinate of the point D can be ascer- 
tained purely from the data as to the curve at C, viz. its height, the rate 
at which this height is increasing (7.e. its slope), the rate at which this 
slope is increasing (2.e. its “curvature” (§ 53)), the rate at which 
this “curvature ”’ is increasing, etc., etc. In fact, the theorem states 
that the ordinate D# is the sum of various magnitudes: first, ¢(a), 
which is represented by 46 (for this is the same as AC); secondly, 


l 
(x — a)o!(a), which is represented by 66! (for = is the slope of the 


52 INFINITESIMAL CALCULUS 


curve at C, and so = ¢/(a), hence 66! = C6 x ¢!/(2)=(x — a) d'(a)); 
ea zal! 
thirdly, Eels A, which is represented by 6/5/’, when 6!’ is reached 
on 


by drawing the curve Cé!’, which has the same curvature as the prin- 
cipal curve CD has at the point C, but retains that “curvature” (with 
respect to the x-axis, see § 53) throughout; that is, we approach D by 
adding successive corrections. 6 is the position D would have had if 
the ordinate of the curve had remained unchanged from C (so that the 
curve would have followed the horizontal C6); 6! is the position D 
would have had if the rate of increase of the ordinate, z.e. the slope 
of the curve, had remained unchanged from C (so that.the curve would 
have followed Cé'); 6! is the position D would have taken if the rate 
of increase of the slope had remained unchanged from C (so that the 
curve would have followed C6!’), etc. 


67. If we take the point & instead of C,so that a=o0, 
Taylor’s theorem reduces to the simple form 


$(2)=6(0) + $v + 20 4 9" 


This is Maclaurin’s Theorem. 


+ etc. 


The student will observe that (0) is by no means itself zero. It is 
simply that particular value of @(«) obtained by putting x =o. Thus, 
if @ (x) is #8 + 2474 117, P(O) is 117. 


68. A second mode of stating Taylor’s Theorem, and one 
often met with, is obtained by denoting the difference of 
abscissas « —a by f, and replacing « by a+A4 (for, if 
x—a=h,x=a-+A), so that 


o 


hh? b'"(a)h? 


o(a+h)=¢(a)+4'(ah+ 2 AO) era a oes 


2 
or, changing our notation from a@ to x, 


(EM =8@)+ HOt HOE Ho, 


where x now refers to the abscissa of C instead of that of D. 


LTAVLOR’S THEOREM 53 


The student will also sometimes see the theorem expressed 
in the same form, but with y employed in place of &. 


69. There are many applications of Taylor’s theorem in 
economics. Cournot in his Principes Mathématiques makes 
frequent use of it, as does Pareto in his Cours a’ économie 
politique. 

When Z is a small quantity, as in some of Cournot’s cases 
of taxation, then the higher powers of % may be neglected, 
and we have the approximate formula 


d(x +h) = o(x)+hg'(x). 
This is assuming that if the interval AZ is very small, the 
point 6! will coincide approximately with D. 


70. It will be observed that an hiatus was indicated 
in the demonstration of Taylor’s Theorem. ‘This means 
that it is not always possible to develop #(x) in the series 
proposed, and that the attempt to do so will give a diverg- 
ing or indeterminate series. 

It is impossible in so elementary a treatise as this to indi- 
cate in what cases Taylor’s Theorem is applicable. The 
subject is one of great difficulty, and some of the most im- 
portant conclusions relating to it have only recently been 
discovered. 


71. To show the application of Taylor’s and Maclaurin’s 
theorems, let us use them to develop the function (@ + x)”, 
assuming it developable. Since #(~)=(@+.2)", 

$!(x)=n(a +x)", 
pl"(a)=n(v — (atx) 


etc: 


54 INFINITESIMAL CALCULUS 


Hence (0) = a", 
POs "aw, 
$!"(0)= n(n — x)a"-4, 


etc. 
Hence 


p(x) = 6(0) + '(0)x + oO i 


a(n — 1)a"*x? 


= @" + na* lx + 
2 e 


a result which we already know by the binomial theorem. 


Again let us develop sin x, assuming it developable. | 


Since o(x)=sine ~(0)=0, 
op (4) = cos x g'(0)= 1, . 
go! (x) =— sin x O70) ==.0) 
op!) =— cos x g'"(0o)=—1. 
etc. etc. 
Hence 
$(2) = 900) + oor +O SO 


PAR eR sae 2. sh 
3! 


Again let us take : 
x—a-+I 
Si LU Pe eat AS ue 
Since > SOkrap han Olej= 1, 
o'(x)=—(* —a4+ 1), ¢'(a)=— 
pl (x)= 24 —a+1), gl'(a)=2, 


PUNK) = 2 se ta aoa 
Hence, by Taylor’s Theorem, 
$(«)=1—(x—2)4 


2(4% — a)? 34 — a) 


21 34 


={- aoa 


iy 


TAYLOR’S THEOREM 55 


72. Among other important uses of Taylor’s and Maclaurin’s theo- 
rems are the evaluations of the fundamental constants e and 7. 
To obtain e, we develop the function e*, 


(x)= &, G(0) = 1; 
$! (x) = e, OO) = is 
$" (x) =e, Bu eOy ey: 
etc. etc. 
Since (x)= $(0)+ $!(0)x + Bevel Oo : 


Caaee 
we have yas OT ope nee 
If, in this equation, we put x = I, we have 
Fa Cat ice Hie SOE a Rl ee a 

7 Mane ake 


from which e may be computed with any required degree of approxi- 
mation. ¢ = 2.71828 --. 
To obtain 7, develop arc tan x. 


o(*) = arc tana, p(0)= 0, 
EO aearmet — #O)SI. 


If x be less than unity, we know by algebra that * 


= yg? + gt — 76 4 .e,, 


e@= 5 
Hence $"(4)=—244+423—-—62°4-., g!'(0)=0, 
PI (x)=— 243-42? —5-6xt+-, P/(O)=— 
PM(H)H2-3-44— 4-5-6238 4+-, — (0) =0, 

P'(4) = 2-3-4—3°4-5-6x° ++, gro)=4+4}, 
etc. etc. 


ee fe ene 


aoa = Out er 1G: x ae, pee ae s Eau 


* It is assumed here, without proof, that the proper conditions as to 
convergence are fulfilled. 


56 


Let x be la | so that arc tan.x, the arc whose tangent is ae 1G) 
/ 


INFINITESIMAL CALCULUS 


Tv 


3 a Wide 


(7.e. an arc of thirty degrees). The preceding equation then becomes: 


Ee 8 a | I I 


6 V3 3(V3)8 5(V3) 


J I I I 
= —] I ———_+ le st} 
Ge Sy 05. 16 ahve 


whence m= 2V3/1~- 4 USen ene +] 


TZU Sea My se 
= 3.14159 --. 


73- EXAMPLES. 


1. Develop (a — x)~? in series of ascending powers of x. 


— — et he 
a 


14, 


. Develop arc sinx. Ans. « +4 — aa eae 


OMT AAP w DW 


. Develop Va — x. 


x? at x8 


Develop. cosa) (its. eles om eee 


2c ALCON 


. Develop log (1 + «). 


Develop a@°*?, 
Develop &*. Ans. 1 -+ 34+ om. ee 


. Develop 4(e* — e-*). 


ay a . 
(3B 12-4. Br 2Hi0 


. Develop cos 2x. 
. Develop e*séc z- 


2 
. Develop log (1+ sinx). Ans. £m 


Develop arc tan x. 
Develop cos (+ + y). 
; ae Pp, 
Ans. COS x — y sin x — Bis a + Vea Xf eee 


Develop tan (+ + y). 


3 ; 
Ans. tanx + ysec’x + 7” seca tan x +7 secka(1 + 3tan2r)+ +. 


tVA EOKRAL CALCULUS 57 


GHAPTER: Vi 
INTEGRAL CALCULUS 


74. We have thus far been occupied with the derivation 
from F of F', F", etc. But it is possible to reverse this 
process, and, given /’", or any other derivative, to pass back 
tone ils FG 

f(x) was called the derivative of F(x); we now name 
f(x) the primitive of F(x). The first process of obtaining 
f' from / is the subject matter of the wfferential calculus, 
of which the preceding chapters have treated. The process 
of obtaining / from /” is the subject matter of the zz/egra/ 
calculus. 


75. In the differential calculus, we saw that the result of 
differentiation was expressed either in the differential quo- 
tient /'(x), or in the differential #'(~)dx. In the integral 
calculus it is customary to employ only the latter form. We 
called #'(x)dx the differential of F(x); we now call F(x) 
the zntegral of F'(x)dx. We obtained /'(x)dx from (x) 
by differentiation. We obtain F(x) from F'(x)dx by inze- 
gration. ‘The symbol of differentiation was @; that of in- 
tegration is J. 

Knowing that ¢(x?)= 2x dx, we may write fp html re 
or again, since ) 

DNC ED MC AY Ls 


58 INFINITESIMAL CALCULUS 


expresses in the most general manner the process of the 
differential calculus, 


[Fax Se 


expresses the process of the integral calculus. Both equa- 
tions state the same fact looked at from opposite directions. 
The former equation reads, “the differential of (x) is 
F'(x)dx”; the latter may be read, * the function-of-which- 
the-differential-is /''(x)dx is /(x),” for the hyphened words 
are what is meant by “ integral of.” 

The simplest form of the above equation is f WX ae 


76. The symbol f was originally a long S, which was the old 


symbol for “sum of” (to-day it is usual to employ the Greek 2 instead). 
Integration was looked upon as summation. ay being the limit of 
Ay, and Ay being a small part of y, the differential dy was conceived of 
as an infinitesimal part of y. An infinite number of dy’s were thought 
of as making up the ». 


77. As a(x*) = 3x* dx, it follows that 


fs x? dx = 2°. 


But dan) Sa as 
hence fs Ce =e ae 


that is, the integral of 3 x? ¢x (or the primitive of 3 x?) may be 
x* or x*+ 5, and evidently also x«* + 17 or x° + any constant 
whatever. In general, if F''(x)dx is F(x) + C, where C is 


any arbitrary constant. For the latter expression differenti- _ 
ated gives the former (§ 27). 
An arbitrary constant (usually denoted by C) must there- 


INTEGRAL CALCULUS 59 


fore always be supplied after integrating any differential to 
obtain the complete integral. 


78. There is no general method of integration known 
corresponding to the general method of differentiation of 
Chapter I. The only way we arrive at the primitive of a 
given function is through our previous knowledge of what 
function differentiated will yield the given function. 


n+1 
79. faxras SS pee ERC) 
nm+t 
n+1 
provided z is not=—1. For the differential of les +C 


n-+1 
is evidently ax"¢x provided 2+ 1 is not zero; Ze. ncaa 
mis not=—TI1. 

The rule, therefore, for integrating the simplest algebraic 
function is to increase the exponent by one, and divide the 
coefficient by the exponent so increased (and then, of 
course, to add an arbitrary constant). 


Thus, i) 2a2dx is 2034. 


80. EXAMPLES. 


fede =? 
sade =? 
Weer a Ans. 42°40, 


x aK _ 
Fi ¢ 
Neg cne 
ax I 
eta ae 
ee Ans. re +C, 
Ax. 


xt 


60 INFINITESIMAL CALCULUS 


81. It may seem at first that a result involving an arbitrary 
constant can be of little use. But this is far from true. 
Though we cannot determine the arbitrary constant from the 
given differential, we may have, in any particular problem, 
information from some other source which will enable us to 
determine it, and often, as we shall see, we do not need 
to determine it at all. We may interpret the constant C 
geometrically by plotting the equation y= F(x#)+ C. To. 
know /''(x)dx or F'(x) is to know the slope of the curve 
for any value of x. But evidently the slope of the curve 
does not determine the curve; since, if the curve were 
shoved up or down without change of form, it would have 
- just the same slope for the same value of x. The constant 
C has to do with the vertical position of the curve. It has 
nothing to do with its form. 

82. We may profitably follow the plan adopted in intro- 
ducing the differential calculus, and begin by considering a 
mechanical and a geometrical application. 

We have seen that, knowing a body falls according to the 


law Sel ae (1) 


we can show that its velocity at any point is 
492 eee (2) 


Suppose, however, we only know that a body acquired 
velocity according to law (2), can we pass back to law (1)? 
As has been said, in the integral calculus it is customary to 
use the differential form to start with. Accordingly, we 


write (2) in the form 
UST ae he 
Integrating, we have 


2 
s=fg2tdt=2 4 C= 16 r+ C. (3) 


INTEGRAL CALCULUS 61 


- 


Now, although equation (2) with which we started does 
not enable us to judge of the value of C, we may evaluate C 
from outside data. 

Thus if we know that s is measured from the point at 
which the body started to fall, we know that when ¢ was zero, 
s must have been zero too. 

Putting s =o and ¢= 0 in (3), we have 


G=-01-- C; 
or Gao, 


After substituting this value of C in (3), the equation 
takes the definite form 
s= 167". 


83. Of course, C is not always zero. In fact, in the above ex- 
ample, we might reckon the distance s of the falling body not from 
the point where it started, but from a point 27 feet above. We then 


know that when 
¢#=0, S= 27. 


Substituting in (3), we have 
27 = Or Ce Or 4 Cie 2774 
and (3) now becomes 
§==1G6 74 27; 


Evidently the value of C depends solely on what origin we use to 
measure s from. 


84. Similarly, if we know the relation between the slope 


ju WE 
of a curve ~ and its abscissa, we can obtain the equation 
bs 


of the curve, except for an arbitrary constant which regu- 
lates the vertical position of the curve. This example is the 
true inverse of the geometrical illustration in the differential 
calculus (§ 12). But for the purpose of the integral calculus 
we prefer another geometrical example. 


62 INFINITESIMAL CALCULUS 


85. Suppose we have (Fig. 10) a plot of y= f(x). Give 
to x an increment Ax, viz. AE or BK, and consider the 
resulting increment not of y, but of the area OABC or 2. 


B 
! 
I 
1 
J 
! 
t 
| 
| 
1 
| 
| 
| 
' 
! 
| 


O ALE IN 


FIG. ro. 


This increment Az of the area is evidently the small area 
ABDE. This small area is the sum of the rectangle 
ABRKE and the tiny triangle 2DX. The area of the rec- 
tangle is the product of its base Ax by its altitude /(x). 
So that 
Az=/(x)Ax+ BDK. (1) 
Evidently the smaller we make Ax, the smaller the area 
of BDK becomes relatively to the small rectangle, and may 
finally be neglected, giving the important equation 


Tbe Ny S G77 hos (2) 


This is not, of course, a mere approximation. It is abso- 
lutely exact. : 


INTEGRAL CALCULUS 63 


The reasoning just given is to be understood as an elliptical form 
of the following: 


Dividing (1) by Ax, we have 


paY oul BDK. 
Re ae eo (3) 


Now is less than 


BDK 
A 

rect HK , He rect 7K | 

Aa ar ome Kass 


But the area of a rectangle divided by its base is its altitude — in 
this case DX. Hence (3) may be written 
¢ 
a = f(*)+ something less than DX. 
bs 


It is evident that when Ax becomes zero, DX becomes zero, and 
“something less than DXA becomes zero,” so that our equation becomes 


as 

ve =f(*); 
which may be written 

as = (ajax. 


This equation is often written 
a2 —=)) 2%, OF z= | ydx, 
y being the usual symbol for /(), the ordinate of a curve. 


86. Suppose y or f(x) to be 
30° +5; 
that is, let y= 3%?+5 be the equation of a curve. The 


integral calculus enables us to obtain the area z in terms of 
the abscissa x. 


We know that dz = (341 +5) dx, 


z= f (32°+ 5) ae, 
g=xeit+5x+C. (1) 


64 INFINITESIMAL CALCULUS 


The student may test the correctness of this integral by 
differentiating it and obtaining (3 «7 + 5) dx. 

It remains to determine C. Since we intended to meas- 
ure the area z from the y-axis, evidently z vanishes when x 
vanishes. Putting x and z both equal to zero in (1), we 
obtain C=o. (If we had measured area from some other 
vertical than the y-axis, the value of C would be different.) 
Hence (1) becomes = 4° + 5. 


Thus suppose «= 3; ithen z= 42. That is, thewareaginciuded 
between the curve vy = 3? + 5, the axes of codrdinates and a vertical 
3 units from the y-axis is 42 units. If the linear units be inches, the 
area units are square inches. 


87. We see more clearly now than in § 76 why integration was first 
conceived of as summation. ‘The area z is evidently the sum of a great 
many Az’s, and at the limit is conceived of as the sum of an indefinite 
number of @2’s. 

The dz is chought of as an elementary strip of area infinitely narrow 
—the limit of ABDL. 


88. The problem of obtaining curvilinear areas was one of the 
earliest and is one of the most important of the applications of the 
integral calculus. Previous to the discovery of this branch of mathe- 
matics only a very few curves, such as the circle and parabola, could 


be so treated. 


89. We are here chiefly interested in the geometrical 
symbolism. We have seen that the slope of a curve is 
the differential quotient of its ordinate (with respect to its 
abscissa). We now see that the ordinaze in turn is th: 
differential quotient of its area (also with respect to the 
abscissa). For d= yd@x means simply 


az 


ay 3 


INTEGRAL CALCULUS 65 


If we wish to make a graphic picture of any function and 
its derivative, we can represent the function either by the 
ordinate y of a curve or by its area, while its derivative will 
then be represented by its slope or ordinate respectively. 

If we are most interested in the /wacton, we usually 
employ the former method (in which the ordinate repre- 
sents the function) ; if in its derivative, the latter (in which 
the ordinate represents the derivative). That is, we usually 
like to use the ov/inaze to represent the main variable under 
consideration. 

Jevons in his Zheory of Political Economy used the 
abscissa x to represent commodity, and the area z to repre- 
sent its total utility, so that its ordinate y represented 
“marginal utility” (ze. the differential quotient of total 
utility with reference to commodity). Auspitz and Lieben, 
on the other hand, in their Untersuchungen tiber die Theorte 
des Preises, represent total utility by the ordinate and margi- 
nal utility by the s/ope of their curve. 


go. The method of integration enables us not only to 
obtain the particular curvilinear area described, but also an 
area between two limits, as 4B and A'S’ (Fig. 10). Evi- 
dently this area is the difference of two areas OA'B'C 


pce U4 Ce ethe: first is thé) value of S Seas, when 
OA! (or x2) is put for x in the integral when found, while 


the second is the value of the same integral for « =OA 
(or x,). This is expressed as follows: 


er 


* Jaya, 


et | 


and is called an zndegral between limits, or a definite integral. 
The reason it is called definite is that it contains no arbi- 


66 INFINITESIMAL CALCULUS 


trary constant, for this constant disappears when one of the 
two integrals concerned is subtracted from the other. 


Thus, if f. S(aax be F(x) +6, 


oe J (x)ax 


means simply (/(*#,)+C)—(7(*)+C), 


which reduces to /'(x,)— (x), for C must be taken to be 
the same in both integrals. 


L=2. 


The area between the curve 3.x? + 5, the x axis, and the two verti- 
cals erected at x = 2 and x = 4 is 


a=4 
ie (342+ 5)dx =[48 + 544 Clraa—[e? + 54+ C]raze = 66, 


for the C drops out, since for each expression the area is measured from 
the same vertical, though no matter waz vertical. 


It is usual to abbreviate the expression for limits. 


x4 4 
Thus, instead of ff J (x)dx, we write af Tiajan 
~—2 2 


g1. There are certain general theorems of integration 
corresponding to the general theorems of differentiation of 
Chapter II. Of these the two most important are: 


; [A a)dx =Kf fade 
and LAG) + AG) $e) Jae 
= {Ade + f A(a)ax + f f(x)ax LES he 


The proof of the first is simple, for the integral of the 
right side of the proposed equation is A(/(«)+C), or 
KF (x)+ KC or KF (x)+C', where /(x) means the primi- 


INTEGRAL CALCULUS 67 


tive of #(x) and C is an arbitrary constant. But C’ might 
as well be written C, since its value is anything we please. 

The integral on the left is also A/A(x)+C; for this 
differentiated gives A/(x)dx. 

The proof of the second is also simple. If we denote 
the primitives of A(x), A(x), ---, by A(x), F(x), +++, it is 
evident that the integral on the right is 

F(x) + C, + a(x) + Cy + Fa(e) + Cz +, 
or F(x) + F(x) + + +0, (1) 
where C is Ci} +C,+C;, and is therefore arbitrary. The 
integral on the left is the same quantity (1), for the differ- 
ential of (1) is (§ 26), 


d( F(x) + F(x) +» +C)= dF,(x) + dFi(x) + 
=f(x)dx + fia)dex ++ = (f(x) + a(x) + Jade. 


Q2. EXAMPLEs, 
1. Integrate (1+ a+ d)x? dx. 
2. Integrate a2? dx + 723 dx + 5 x dx. 
3. Integrate (4 + 2) cr! dx + kx dx. 
Ans, (h +2) fess +2 a+] : 
4. If the velocity of a body increases with the time according to the 


formula —_ 32°, find the formula for the distance traversed. 


5. How far does it move between the instant when ¢ is 3 seconds 
and that when ¢ is 5 seconds? | 
6. Find the expression for the area (corresponding to z in Fig. 10) 
for the curve whose equation is y=54?+ 2. Ans. tid +2x44+C. 
7. What is the value of that area for the point where x is 1? 
Where x is 3? Where y is 22? 
8. What is the area between the curve, the x-axis, and the two 
verticals erected at x= 2andx—4? Ans. 100, 
9. Solve the same problems for the curve y= +3414; for y= 
wes otor j4 = 4 ax. 
10. Find the area z, for y=a*; y=log (+ +5); y=sinw. 


Ans. te: («+ 5) log(#+5)—*#+C; —cos*#4C. 


68 INFINITESIMAL CALCULUS 


93. Just as we may differentiate successively, so we may 
integrate successively. 
If we perform the integration 


fi J (x)dx and obtain f(x), 


we may then take 


fi A(«)ae and obtain £(x), 
and then if; Ji(x)dx and obtain f(*), 


CLC; etc. 


Instead of writing ff J\(x)dx, we may substitute for f(x) 
its value ye (x)d@x, and we shall have 


f [Sear Jat, 
which, however, is usually abbreviated to if J (x)dx dx, or 
even to Jf PECAN Lee 


Similarly, we may write 


Sff7@ ax ax ax, or SS Ivo dx, ete. 


We may express the double, triple, etc., definite integrals 
also. The full form for the double definite integral would be 


So Se ree jess 


which, however, may be condensed to 


if f f(x) fib 


INTEGRAL CALCULUS 69 


94. To apply these ideas we recur to our old example of a falling 
body. Suppose our first knowledge is not s= 167? nor a= 324, but 


WA 
(see 32; that is, we simply know that the acceleration is a given con- 


at? 
stant (32 velos per sec.), or to be more general let us call this con- 
stant s. ad ( a 
; areata “Saas at 
The given equation, 2 = g, means, as we know, =10, OF 
ds 
ad{|—)=¢2 ad, 
(G)=e% 
: : ads 
whence, integrating, wi gt+C; (1) 
but this may be written as = gtdt+ Cd, 
whence, integrating again, s=4tot?+ t+ kK. (2) 


We have still to determine the arbitrary constants Cand X. If the 
distance s is measured from the starting-point, then s and ¢ vanish 
simultaneously. Substituting zero for them both in (2), we obtain 

e 
am, 

It remains to determine C. 

To do this we take equation (1) and suppose the body falls, not 
from rest, but with an initial velocity of w feet per second; then when 
¢ is zero, ge is wt, 

at 
and (1) then reduces to 
OG fore. C— tu. 

Substituting C= « and K =0 in equation (2), we have 

s=tel + ut, 
the general equation of falling bodies. 


Q5- The process which we have followed out in detail from the 


equation 
ase) 
at? 
may be condensed as follows: 


$ Si\ ee =( fea, 
= (+ C)de 


70 INFINITESIMAL CALCULUS 


96. The simple transcendental integrals are obtained as follows : 

Since d(sinx)= cosx dx, then § cosx dx =sinx + C, 

Since d(cos x) = — sin x dx, then j- sinxdx = cos x+C, 
whence fain («)adx =—cosx — C=—cosx+4 C, 


for C is perfectly arbitrary. 
Since!) | id(as) 5 eee then f a*Logadr _j24 ¢ 
Loge L 


oge 
whence fe Fy a* Loge + C. 
Loga 
Also fe ac Tea te + C. 
loga 
Since d@arc sinx ASSES then nee arcsine + C. 
1 — x2 Vi — x? 


-, then i\ ai 5 = are tan & + Cc. 


Since darctanr« = ax 
I I+24 


gE 
‘ ax ax ’ 
Since dlogx =—, then je =logx+UC 
es Os 
= logx + log K = log (Kx) 
ior Cand X are wholly arbitrary. 


97. We may summarize the formule for integration which 
have been given : 


fade =ax+6, 


axnth ‘ 
fae ds = : +C (when z is not =—1), 


fax ax =alogx+C, 
fie hep (iano AO vy 
Loga 
ka* 


Sal Sic i? 


INTEGRAL CALCULUS 71 


fede=et G; 
lg oe arc tanx + C, 
ax 


—_—__—_ = arc sinx + C, 


Vi ie 
fsine de =—cosx +6, 


ficos em sine iG, 


98. Treatises on the integral calculus are usually ery bulky, be- 
cause they are occupied with the determination of special integrals, 
both definite and indefinite, and with special devices for obtaining 
them. In this little book, which is devoted to only the most general 
and fundamental principles, we may fitly close our discussion at this 
point. Practically, even advanced students of the Calculus usually 
depend on tables of integrals. The reader is referred to B. O. Pierce’s 
“Short Table of Integrals.” Completer tables occupy large quarto 
volumes. An absolutely complete table does not exist, for there are 
multitudes of integrals which have never yet been solved. 


99. We may, however, point out one tool for integrating 
already in the reader’s possession. 
Suppose we have to integrate 


Retrial) aay 
This may evidently be put in the form 
(x? + 2)°x dx, 


or d(x’? + 2)? 2x dx, 
or £ (x? + 2)°d (x), 
or 4 (x? + 2)°a (x? + 2), 


and in this form it is easily integrated. 


72 INFINITESIMAL CALCULUS 


For, putting # = x* + 2, we have 


4 du, 
the integral of which is 
u's 
—+C 
8 nee 
2 4 
a bs — 4C 


This device consists in changing the variable, getting rid 
of dx, and obtaining instead a differential of some other 
variable, ~, in terms of which the whole expression may be 


written. 


IOO. EXAMPLES. 


8 
1. Eee PY catetreat! 
; Va? + bx? 
ae : 2.5 2a 
2.  Vadx=? Ans. p35 VO + ba% 
les 
GOS arse ee ata 
x3 2x4 
8. ( es, 
a — bx* 
(y=? 
5 (aed : 9, f (a+ 3.22)8 de. 
“J (ti ix4)3 


—2axr 
10. if V4 — x2 x2 ° 


2 
Ans. @fx+3 mers x 


APPENDIX 73 


APPENDIX 
FUNCTIONS OF MORE THAN ONE VARIABLE 


to1. We have had to do hitherto with functions of only 
one variable, such as x*+2%-+ 3. But the magnitude 
x* + 2xy-+ 3”, for instance, is dependent for its value on 
two variables, x and y; z.e. is a function of x and y. 

The relation z=.x«’?+2xy+3,y’, or, more generally, 
z= (x, y), states that z is a function of « and y; that is, 
that a change either in x or y produces a change in 2. 


Thus, the speed of a sailing vessel is a function of the strength of 
the wind and the angle at which she sails to the wind. 

The force which produces tides is a function of the earth’s distance 
from the moon and its distance from the sun. 

The price of stocks is a function of the rate of dividends and of the 
rate of interest. 


Similarly, z = /(x, y, 2) expresses the fact that z« de- 
pends on x, y, and z, and so on for any number of variables. 


Thus, the force which guides the moon is a function of its distance 
from the earth, its distance from the sun, and the angle between the 
directions of these two distances. 

The price of a Turkish rug is a function of the prices of its constitu- 
ents, the cost of transportation, the rate of tariff, etc. 


If for w= F(x, y, 2), the condition of some special 
problem should require z to remain constant, the function 
may be written as w = (x, y) ; and if y is also constant, as 


W(x). 


76 INFINITESIMAL CALCULUS 


Thus, the speed of a sailing vessel is a function of her angle to the 
wind, if the strength of the wind remain constant. 

The price of woollen cloth is a function of the price of wool, if the 
cost of labor, etc., remain constant. 


102. Since the terms of an equation can be transposed, 
it is always possible to gather them all on the left side, thus 
reducing the right side to zero. y=V«x?+1 is the same 
equation as y*»—«?—1=0. The left member is here a 
function of x and y. -And in general it is evident that any 
relation between two variables y= F(x) can be reduced to 
the form #(x, y)=0. When expressed in the first form, y 
is called an exfécit function of x. In the latter it is an 
zmplicit function of x. 

In like manner, any relation z=/(x, y) can be reduced 
to the form $(%, y, 2)=0; any relation w= (4, y, 2) to 
o(x, y, 2, w)=0, and so on. 


103. We have seen that $(%, ))\= 0 .0rey,— Agen 
always be represented by a curve with x and y as the two 
coordinates. So, also, $(%, y, 2)=0 or z= F(x, y) can 
always be represented by a suzface with x, y, and z as the 
three coordinates. 

Draw three axes at right angles to each other, such as the 
three edges of a room, meeting at a corner on the floor, the 
x-axis being directed, say, easterly, the y-axis northerly, and 
the z-axis upward. 


To represent s= x? + 2axy+ 37, 
let « have any particular value, such as 2, and y, 1. 
Then hs Se Di year Jia Om wae Sat elm 


Find the point in the room which is 2 units east of the 
corner, 1 unit north of it, and 11 units above it. This is 


APPENDIX 75 


one point of the required surface. By taking all possible 
combinations of values of x and y, and finding the result- 
ing values of z, we can find aZ points on the surface. 


104. When z= F(x, y), we may vary x by Ax, while y 
remains constant, and thus cause in Z an increment denoted 
by Az. The ultimate ratio of Az to Ax is expressed by 
0 OF (x 

BLES) 


+ 


Ox Ox 


and is called the partial derivahve of F(x, y) with respect 

to x. 

os OF (x, y) 

Ne a 

is the partial derivative with respect to y; z.e. the derivative 

obtained by keeping x constant during the differentiation. 
Observe that the symbol 0, denoting partial differentia- 

tion, is not identical with @. 


Similarly, 


105. The geometrical interpretation of these partial deriv- 
atives can be made evident. If on the surface, z= F(x, y), 
say the surface of a stiff felt hat, we take any given point P 
and pass through it a vertical east and west plane, the plane 
and surface intersect in a curve passing through 2 ‘The 
tangential slope of this curve at P (or, as we may Call it, the 


E-W slope of the surface itself) is a. For the coordi- 
x 


nates of P are x, y, 2, and those of a neighboring point Q 
on the curve (and therefore on the surface) are x + Ax, ), 
z+ Az, where Ax is the difference between the «x’s of P 
and Q,and Az the difference between the 2’s ; the y’s are by 
hypothesis the same. ‘The slope of the line joining Pand Qis 


76 INFINITESIMAL CALCULUS 


aay and its limiting value, lim ae or id is the slope of 
Ax Ax Ox 


the curve at P (see § 12); z.e. the E—W slope of the sur- 


face. 

Similarly, te or eedG is the north and south slope of 
the surface. ‘ 

These two primary slopes of the surface can be repre- 
_ sented by placing two straight wires or knitting needles 
tangent to the hat at the point % one in an E—W vertical 
plane and the other in a N-S vertical plane. 

If we take azy neighboring point # on the surface, its 
coordinates are x + Ax, y+ Ay, z + Az, where the A’s are 


the differences of coordinates of P and A. 


q0in /vand 9) chen = represents, not the true slope 
x 


of the line PR, but its east and west slope (not, of course, the 
east and west slope of the surface itself). It is the rate the 
line ascends in comparison, not with its true horizontal prog- 
ress, but with its eastward progress. A climber ascending a 
northeasterly ridge may be rising 5 feet for every 3 of hori- 
zontal progress, but yet rising 5 feet for every 2 of eastward 
progress. We have to do with the latter rate, not the former. 


So also 2 is the north and south slope of the same line PR. 


Now let & approach P (along any route whatever upon 
the surface) until it coincides. ‘The line PP approaches a 
limiting position which is a new tangent to the surface (a 
tangent to that curve in the surface which & traced in ap- 


proaching P?). The E-W slope of this tangent is lim = j 
called a and its N-S slope, ge 
ax dy 


Representing this tangent by a third wire, we have three 


APPENDIX 77 


tangent wires through 7, one in an E-W vertical plane, a 
second in a N-S vertical plane, and the third, anxy other 


tangent. The first has no N-S slope; its E—W slope is 


pee The second has no E—W slope; its N-S slope is “ . 


tbe az az 
The third has both kinds of slope, viz.. — and —- 
ax dy 
106. As will be shown, the relation between these various 
derivatives is 


dz =F ay 4 gy, (1) 
Ox oy 


which may be thrown into the forms: 


Ge _ 92 | OB ay 
Qe 0x) Oye ax 


or ; (2) 
G2. OB. Bes) 02 


dy dx dy ay 
The form (1) has the great advantage of symmetry. It 
seems, however, to conceal the existence of Be or ei) which 
18 EES 
are brought out in (2). These last two magnitudes require 


merely a word of explanation. is not an upward slope 
Ke 


at all, as it does not involve the vertical z. It is the incli- 
nation of the third wire across the floor, the rate at which 
a moving point on it proceeds north in relation to its east- 
ward progress. 


107. The proof of the formula stated in the last section is as 
follows : * 


* In order to master and remember this proof, the student is advised 
to construct for it some actual physical model. He will then find it 
extremely simple. 


78 INFINITESIMAL CALCULUS 


We first assume that all wires through P tangent to the surface lie 
in one and the same plane called the ¢angent plane. This assumption 
is analogous to that in § 14, that the progressive and regressive tan- 
gents coincide. There is an exception if the surface has an edge or 
wrinkle at the given point. 

Let us take in this plane the three tangent wires above considered, 
viz. the two primary wires (in vertical planes running E-W and N-S 
respectively) and the wire obtained as the limiting position of PQ. 
Take a point Q/ on this third or “ general” wire, having codrdinates 
x + Alx, y + Aly, 2+ Al'z. (The primes serve to distinguish Q’ on 
the tangent plane from Q on the surface.) 

Through Q! pass two vertical planes running E-W and N-S respec- 
tively. We already have two such planes through ?. These four 
vertical planes cut the tangent plane in a parallelogram, of which PQ! 
is a diagonal and the “ primary wires” are the two sides meeting at 
P. Denote the two vertices as yet unlettered by 4 and A, the former 
being in the E-W and the latter in the N-S primary wire. 

A'z being the difference in level of P and Q! is the sum of the dif- 
ference in level of P and H and of H and Q, just as the difference in 
level between Mount Blanc and the sea is the sum of the elevation 
of Lake Lucerne above the sea and of Mount Blanc above the Lake. 
(It does not matter whether / is or is not intermediate in level between 
P and Q’, for if not, one of the heights considered becomes negative.) 

Now the difference in level of P and / is 


oz Ax 


Ox 


for the difference of level, 2, between any two points, as JZ and VV 


iN 


h 


a 
Fic. rx. 


(Fig. 11) is the product of the siope of AZM by the horizontal interval, 


a, between them (since: slope of AZV =—, whence 4=a xX slope of 
@ 


APPENDIX fee, 


MN). 3 is known to be the slope of PQ’, and alx is the E-W 
Pe 


interval between P and Q’, and therefore also the E-W interval (or 
in this case the horizontal interval) between P and / (since H and Q 
are in the same N-S plane). 

Again the difference in level between Hand Q?! is 


Oz ! 
— Ay. 
oy “A 
For i being the slope of PA, is also the slope of YQ! parallel to 
y 
PK, and A'y, being the N-S interval between P and Q’, is also the 


N-S (and in this case horizontal) interval between 4 and Q! (since 
ff and FP are in the same E-W plane). 


Therefore, 
j A!z = 8% pty 4 9% aly, (1)! 
x 0 
which is the prototype of the desired result (1). 
: NE. Oe Ose PAL (2)! 
This may be written I NF Sexe 
! 
Now ze is the E-W slope of the “general tangent” wire PQ’. 
me ! 
But we have seen that @2 6 also this slope. Again, 4 is the inclina- 
ax A'x 


tion of this same wire across the floor (the rate at which a point 
moving on the wire proceeds xorthward relatively to its eastward 


progress). But so also is & (§ 106). Substituting therefore these 
és 


values for the primed expressions, we have 


LEX AOV AIX 


which may be thrown into the form 


Oz Oz 
ag = — dx. — dy. 
Ox ay” 
In this, dz is called the ‘otal differential of z, while 2 ax and Oo dy 
are its partial differentials. ox oy 
It is evident that we should reach the same result if in the preced- 
ing reasoning we had employed & in the way we did employ Z, and 


80 INFINITESIMAL CALCULUS 


vice versa; also that we could have divided (1)! by A'y instead of 
by Alx. 


108. The formula (1) (§ 106), or its two alternative 
forms (2), enable us to ascertain the direction of amy tan- 
gent line to a surface. 

Thus, let the surface be 

2=x2?+ 2ay74+ 37%, 
and let it be required to determine any tangent line at the point whose 
x and y are I and I respectively; z is evidently 6. 
1. The primary E-W tangent wire at this point has an E-W slope 


Li + 2y = 4, found by differentiating the above equation treat- 


Ox 


ing y as constant, and has no N-S slope. 


2. The primary N-S tangent wire at this point has a N-S slope 
) 


= =2x-+ 6y = 8, and has no E-W slope. 
Ly 

3. The tangent wire in the vertical plane running northeast and 
southwest has an E-W slope of 
as ELLOS: Oz ; dy 
dx OLR Ovi rax 


= ge 
A ax 


=A) 0 ¢ Lice 2, 
and a N-S slope of 
Enos! _ 1 08 


as Ox ay oy 


4. The tangent wire in the vertical plane running northwest and 
southeast has the two slopes 
4+ 8(—1)=—4 
and 4(—1)+8=+4. 
5. The tangent wire in the vertical plane cutting between north and 
east so as to be advancing north twice as fast as east 


( ee ay — 2), 
Ux 


APPENDIX 81 


dz _ 02 , Oz | w& 

CAEL OV: ax 
=4+8 X 2 = 20, 
_ 02 2%, OF 

a Ox aye voy 
=4X 5+8= 10, 


and so on for any tangent wire whatever. 


has slopes of 


and 


T0Q. EXAMPLES. 

1. Find the slopes of the five sorts above indicated for the same 
surface at the point for which x = 3 and y= 2. 

2. At the point where x =—1I, y=—I. 

SaeAtine pointwhere + —\0,) = O, 

4. For the surface z= a3+ 2427+ e+ayt+y+ 3%+ 7° at the 
Denies) 0,9.) 2-21, 


5. For the surface 
= xty— 2x4 4+ 3 
Beene pointur — 2, 19 =) 3. 
6. On the same surface at the same point, what are the E-W and. 


N-S slopes of the tangent line which progresses northward 3 times 
as fast as eastward ? 4 times ? 3} times? 


7. Answer the same questions for z= log y + 3% + xy. 


110. When we have a function of more than two vari- 
ables, asw= (x, y, 2), there is no mode of geometrical 
interpretation corresponding to the curve for y= (x) and 
surface for z= (x, y) (unless, indeed, we posit a “ fourth 
dimension,” and speak of a “‘ curved space’’ of three dimen- 
sions whose coordinates are x, y, 2, w!). 

It may be shown, however, in a manner strictly analogous 
to the process of § 107, but without employing the geomet- 
rical image, that 

ae _ Ow 
Ox 


ax +S te 


82 ; INFINITESIMAL CALCULUS 


This differential equation is elliptical for the three equations 
obtained by dividing through by dx, dy, and dz. 

The theorem and its proof are extensible to any number 
of variables. 


Imm. A very important application of the principle of 
partial derivatives occurs when we have but two variables, 
but y is an implicit function of x; ze. when (x, vy) =o. 


We are enabled to obtain the derivative 2s without being 
by 


obliged first to transform the implicit function into the - 
explicit form y= F(x). 


Thus, if +2 + y? = 25, we may find Bs without changing the equa- 
tion to the form y=+V25 — 22. a 


112. We know from § 106 (2) that if z= (x, y), then 


dz _ 9 (x,y) | OP(%, 9) 
ax Ox oy ax 
which may also be written in two other forms, as given in 


§ 106. 
When 2 is zero, as in the case now being considered, then 


a 


o is also zero (§ 27, end). Making this substitution in the 
be 


above equation, we obtain 


Op(x, ¥) 


ay oy, Ox 
dx Oba, y) 
oy 


In words: Zo find the differential quotient of y with re- 
spect to x when the functional dependence between x and y is 
expressed in the implicit form (x, y)=0, differentiate the 
Junction (x, y) with respect to x, treating y as constant, 
and then again with respect to y, treating x as constant. 


APPENDIX 83 


Take the partial derivative found from the first differentia- 
tion, divide tt by that found from the second, and prefix the 
minus Sign. i 


dy 
ax 


as 


Thus, if 427+ y2= 25, or «2+ y?— 25 =0, we may find 
follows : 

The partial derivative of «2 + y? — 25 with respect to x is 2.x, and 
with respect to y, 2y. Hence 


SE acs Nee oy 
ax 2y y 


This result is expressed in terms of both x and y, but it may be 
transformed so as to involve but one variable. Thus, substitute for y its 
value as obtained from +? + y? = 25, viz. +V25 —«*. Then 


ad Aelia Ma NS OL 
ax + V25 — x2 


a result identical with that obtained by differentiating the explicit form 


yor V 25 — x7, 


II3. EXAMPLES. 


1. Find 2, if xy =1. 
ax 


2. Find %, if 2224 337-4=0. 
ax 


dy 


3. Find -—, 
ax 


if axy + dx3y2 = o, 


Ai Mind orf e —o, 
DEE Kime PCy BOR 


5. Find Oe if cos(4y)= x. 
ax 
6. Find a, if log(#*y?) + 28 + 84+ 2x%y+a=0. 
as 


7. Show § 112 geometrically, 


84 INFINITESIMAL CALCULUS 


114. Functions of many variables are peculiarly appli- 
cable in economic theory, though as yet they have been 
very little employed.* Many fallacies have been committed 
from lack of this more general conception of functional de- 
pendence, and from the tacit assumption that mere curves 
are capable of delineating any sort of quantitative relation. 
This is an error only one degree less flagrant than the errors 
of those whose sole mathematical idea is that of the con- 
stant quantity. 


* See, however, Edgeworth’s J/athematical Psychics, 1881; the 
author’s Mathematical Investigations in the Theory of Value and 
Prices, 1892; and Pareto’s Cours d’économie politique, 1896-7. 


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UNIVERSITY OF ILLINOIS-URBANA 


515F53B1906 c001 
A BRIEF INTRODUCTION TO THE INFINITESIMA 


7 


NOY 


0172 


